# Mie scattering

The Mie solution to Maxwell's equations (also known as the Lorenz–Mie solution, the Lorenz–Mie–Debye solution or Mie scattering) describes the scattering of an electromagnetic plane wave by a homogeneous sphere. The solution takes the form of an infinite series of spherical multipole partial waves. It is named after Gustav Mie.

The term Mie solution is also used for solutions of Maxwell's equations for scattering by stratified spheres or by infinite cylinders, or other geometries where one can write separate equations for the radial and angular dependence of solutions. The term Mie theory is sometimes used for this collection of solutions and methods; it does not refer to an independent physical theory or law. More broadly, "Mie scattering" suggests situations where the size of the scattering particles is comparable to the wavelength of the light, rather than much smaller or much larger.

Mie scattering (sometimes referred to as a non-molecular or aerosol particle scattering) takes place in the lower 4.5 km of the atmosphere, where there may be many essentially spherical particles present with diameters approximately equal to the size of the wavelength of the incident light. Mie scattering theory has no upper size limitation, and converges to the limit of geometric optics for large particles.[1]

## Introduction

A modern formulation of the Mie solution to the scattering problem on a sphere can be found in many books, e.g., J. A. Stratton's Electromagnetic Theory.[2] 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.

For particles much larger or much smaller than the wavelength of the scattered light there are simple and accurate approximations that suffice to describe the behaviour of the system. But for objects whose size is similar to the wavelength, e.g., water droplets in the atmosphere, latex particles in paint, droplets in emulsions, including milk, and biological cells and cellular components, a more detailed approach is necessary.[3]

The Mie solution[4] is named after its developer, German physicist Gustav Mie. Danish physicist Ludvig Lorenz and others independently developed the theory of electromagnetic plane wave scattering by a dielectric sphere.

The formalism allows the calculation of the electric and magnetic fields inside and outside a spherical object and is generally used to calculate either how much light is scattered (the total optical cross section), or where it goes (the form factor). The notable features of these results are the Mie resonances, sizes that scatter particularly strongly or weakly.[5] This is in contrast to Rayleigh scattering for small particles and Rayleigh–Gans–Debye scattering (after Lord Rayleigh, Richard Gans and Peter Debye) for large particles. The existence of resonances and other features of Mie scattering make it a particularly useful formalism when using scattered light to measure particle size.

## Computational codes

Mie solutions are implemented in a number of programs written in different computer languages such as Fortran, MATLAB, and Mathematica. These solutions solve for an infinite series, and provide as output the calculation of the scattering phase function, extinction, scattering, and absorption efficiencies, and other parameters such as asymmetry parameters or radiation torque. Current usage of the term "Mie solution" indicates a series approximation to a solution of Maxwell's equations. There are several known objects that allow such a solution: spheres, concentric spheres, infinite cylinders, clusters of spheres and clusters of cylinders. There are also known series solutions for scattering by ellipsoidal particles. A list of codes implementing these specialized solutions is provided in the following:

A generalization that allows a treatment of more generally shaped particles is the T-matrix method, which also relies on a series approximation to solutions of Maxwell's equations.

## Approximations

### Rayleigh approximation (scattering)

Rayleigh scattering describes the elastic scattering of light by spheres that are much smaller than the wavelength of light. The intensity I of the scattered radiation is given by

${\displaystyle I=I_{0}\left({\frac {1+\cos ^{2}\theta }{2R^{2}}}\right)\left({\frac {2\pi }{\lambda }}\right)^{4}\left({\frac {n^{2}-1}{n^{2}+2}}\right)^{2}\left({\frac {d}{2}}\right)^{6},}$

where I0 is the light intensity before the interaction with the particle, R is the distance between the particle and the observer, θ is the scattering angle, n is the refractive index of the particle, and d is the diameter of the particle.

It can be seen from the above equation that Rayleigh scattering is strongly dependent upon the size of the particle and the wavelengths. The intensity of the Rayleigh scattered radiation increases rapidly as the ratio of particle size to wavelength increases. Furthermore, the intensity of Rayleigh scattered radiation is identical in the forward and reverse directions.

The Rayleigh scattering model breaks down when the particle size becomes larger than around 10% of the wavelength of the incident radiation. In the case of particles with dimensions greater than this, Mie's scattering model can be used to find the intensity of the scattered radiation. The intensity of Mie scattered radiation is given by the summation of an infinite series of terms rather than by a simple mathematical expression. It can be shown, however, that scattering in this range of particle sizes differs from Rayleigh scattering in several respects: it is roughly independent of wavelength and it is larger in the forward direction than in the reverse direction. The greater the particle size, the more of the light is scattered in the forward direction.

The blue colour of the sky results from Rayleigh scattering, as the size of the gas particles in the atmosphere is much smaller than the wavelength of visible light. Rayleigh scattering is much greater for blue light than for other colours due to its shorter wavelength. As sunlight passes through the atmosphere, its blue component is Rayleigh scattered strongly by atmospheric gases but the longer wavelength (e.g. red/yellow) components are not. The sunlight arriving directly from the Sun therefore appears to be slightly yellow, while the light scattered through rest of the sky appears blue. During sunrises and sunsets, the effect of Rayleigh scattering on the spectrum of the transmitted light is much greater due to the greater distance the light rays have to travel through the high-density air near the Earth's surface.

In contrast, the water droplets that make up clouds are of a comparable size to the wavelengths in visible light, and the scattering is described by Mie's model rather than that of Rayleigh. Here, all wavelengths of visible light are scattered approximately identically, and the clouds therefore appear to be white or grey.

### Rayleigh–Gans approximation

The Rayleigh–Gans approximation is an approximate solution to light scattering when the relative refractive index of the particle is close to unity, and its size is much smaller in comparison to the wavelength of light divided by |n  1|, where n is the refractive index.

### Anomalous diffraction approximation of van de Hulst

The anomalous diffraction approximation is valid for large (compared to wavelength) and optically soft spheres. (Soft in the context of optics implies that the refractive index of the particle (m) differs only slightly from the refractive index of the environment, in comparison with scattering of acoustic waves, and the particle subjects the wave to only a small phase shift.) The extinction efficiency in this approximation is given by

${\displaystyle Q=2-{\frac {4}{p}}\sin p+{\frac {4}{p^{2}}}(1-\cos p),}$

where Q is the efficiency factor of scattering, which is defined as the ratio of the scattering cross-section and geometrical cross-section πa2.

The term p = 4πa(n − 1)/λ has as its physical meaning the phase delay of the wave passing through the centre of the sphere, where a is the sphere radius, n is the ratio of refractive indices inside and outside of the sphere, and λ the wavelength of the light.

This set of equations was first described by van de Hulst in (1957).[5]

## Mathematics of Mie scattering

Values commonly calculated using Mie theory include efficiency coefficients for extinction ${\displaystyle Q_{e}}$, scattering ${\displaystyle Q_{s}}$, and absorption ${\displaystyle Q_{a}}$.[6][7] These efficiency coefficients are ratios of the cross section of the respective process, ${\displaystyle \sigma _{i}}$, to the particle protected area, ${\displaystyle Q_{i}={\frac {\sigma _{i}}{\pi r^{2}}}}$, where r is the particle radius. According to the definition of extinction,

${\displaystyle \sigma _{e}=\sigma _{s}+\sigma _{a}}$ and ${\displaystyle Q_{e}=Q_{s}+Q_{a}}$.

The scattering and extinction coefficients can be represented as the infinite series:

${\displaystyle Q_{s}={\frac {2}{x^{2}}}\sum _{L=1}^{\infty }(2L+1)(|a_{L}|^{2}+|b_{L}|^{2})}$
${\displaystyle Q_{e}={\frac {2}{x^{2}}}\sum _{L=1}^{\infty }(2L+1)Re(a_{L}+b_{L})}$

where ${\displaystyle x=2\pi r/\lambda }$ is the dimensionless diffraction parameter, Re is the real part of the sum of complex numbers, and ${\displaystyle a_{L}}$ and ${\displaystyle b_{L}}$ are expansion coefficients

${\displaystyle a_{L}={\frac {S_{L}(x){\frac {S_{L}'(mx)}{S_{L}(mx)}}-mS_{L}'(x)}{C_{L}(x){\frac {S_{L}'(mx)}{S_{L}(mx)}}-mC_{L}'(x)}}}$
${\displaystyle b_{L}={\frac {mS_{L}(x){\frac {S_{L}'(mx)}{S_{L}(mx)}}-S_{L}'(x)}{mC_{L}(x){\frac {S_{L}'(mx)}{S_{L}(mx)}}-C_{L}'(x)}}}$

expressed in terms of Riccati-Bessel functions ${\displaystyle S_{L},C_{L}}$ that can be expressed in terms of spherical Bessel functions ${\displaystyle j_{L},y_{L}}$ or Bessel functions of non-integer order ${\displaystyle J_{L+1/2},Y_{L+1/2}}$ and obey recurrent relations

${\displaystyle S_{L}(z)=zj_{L}(z)={\sqrt {\frac {\pi z}{2}}}J_{L+1/2}(z)}$
${\displaystyle C_{L}(z)=-zy_{L}(z)=-{\sqrt {\frac {\pi z}{2}}}Y_{L+1/2}(z)}$
${\displaystyle S_{L}'(z)=S_{L-1}(z)-{\frac {L}{z}}S_{L}(z)}$
${\displaystyle C_{L}'(z)=C_{L-1}(z)-{\frac {L}{z}}C_{L}(z)}$

where z can be either x or mx and m is the complex refractive index of the particle relative to the surrounding medium. It has been proposed that the series need not continue past ${\displaystyle L_{max}=x+4x^{1/3}+2}$ for sufficient convergence

Accounting for angle and polarization dependence of scattering, scattering intensity ${\displaystyle I(\theta )}$ and polarization ${\displaystyle P(\theta )}$ are

${\displaystyle I(\theta )=I_{0}{\frac {\lambda ^{2}}{8\pi ^{2}r^{2}}}(i_{1}(\theta )+i_{2}(\theta ))}$
${\displaystyle P(\theta )={\frac {i_{1}(\theta )-i_{2}(\theta )}{i_{1}(\theta )+i_{2}(\theta )}}}$

where i1 and i2 are also intensities in perpendicular polarizations, the square of their respective complex scattering amplitudes

${\displaystyle i_{1}(\theta )=|S_{1}(\theta )|^{2}}$
${\displaystyle i_{2}(\theta )=|S_{2}(\theta )|^{2}}$.

These scattering amplitudes are similarly represented as a sum of expansion coefficients and scattering angle functions ${\displaystyle \pi _{L},\tau _{L}}$ represented in terms of an associated Legendre polynomial (and its derivative)

${\displaystyle S_{1}(\theta )=\sum _{L=1}^{\infty }{\frac {2L+1}{L(L+1)}}(a_{L}\pi _{L}+b_{L}\tau _{L})}$
${\displaystyle S_{2}(\theta )=\sum _{L=1}^{\infty }{\frac {2L+1}{L(L+1)}}(a_{L}\tau _{L}+b_{L}\pi _{L})}$
${\displaystyle \pi _{L}={\frac {P_{L}^{(1)}(\cos \theta )}{\sin \theta }}}$
${\displaystyle \tau _{L}={\frac {d}{d\theta }}P_{L}^{(1)}(\cos \theta )}$.

## Applications

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

### Atmospheric science

Mie scattering occurs when the diameters of atmospheric particulates are similar to the wavelengths of the scattered light. Dust, pollen, smoke and microscopic water droplets that form clouds are common causes of Mie scattering. Mie scattering occurs mostly in the lower portions of the atmosphere, where larger particles are more abundant, and dominates in cloudy conditions.

### Cancer detection and screening

Mie theory has been used to determine whether scattered light from tissue corresponds to healthy or cancerous cell nuclei using angle-resolved low-coherence interferometry.

### Magnetic particles

A number of unusual electromagnetic scattering effects occur for magnetic spheres. When the relative permittivity equals the permeability, the back-scatter gain is zero. Also, the scattered radiation is polarized in the same sense as the incident radiation. In the small-particle (or long-wavelength) limit, conditions can occur for zero forward scatter, for complete polarization of scattered radiation in other directions, and for asymmetry of forward scatter to backscatter. The special case in the small-particle limit provides interesting special instances of complete polarization and forward-scatter-to-backscatter asymmetry.[8]

### Metamaterial

Mie theory has been used to design metamaterials. They usually consist of three-dimensional composites of metal or non-metallic inclusions periodically or randomly embedded in a low-permittivity matrix. In such a scheme, the negative constitutive parameters are designed to appear around the Mie resonances of the inclusions: the negative effective permittivity is designed around the resonance of the Mie electric dipole scattering coefficient, whereas negative effective permeability is designed around the resonance of the Mie magnetic dipole scattering coefficient, and doubly negative material (DNG) is designed around the overlap of resonances of Mie electric and magnetic dipole scattering coefficients. The particle usually have the following combinations:

1. one set of magnetodielectric particles with values of relative permittivity and permeability much greater than one and close to each other;
2. two different dielectric particles with equal permittivity but different size;
3. two different dielectric particles with equal size but different permittivity.

In theory, the particles analyzed by Mie theory are commonly spherical but, in practice, particles are usually fabricated as cubes or cylinders for ease of fabrication. To meet the criteria of homogenization, which may be stated in the form that the lattice constant is much smaller than the operating wavelength, the relative permittivity of the dielectric particles should be much greater than 1, e.g. ${\displaystyle \varepsilon _{\text{r}}>78(38)}$ to achieve negative effective permittivity (permeability).[9][10][11]

### Particle sizing

Mie theory is often applied in laser diffraction analysis to inspect the particle sizing effect.[12] While early computers in the 1970s were only able to compute diffraction data with the more simple Fraunhofer approximation, Mie is widely used since the 1990s and officially recommended for particles below 50 micrometers in guideline ISO 13321:2009.[13]

Mie theory has been used in the detection of oil concentration in polluted water.[14][15]

Mie scattering is the primary method of sizing single sonoluminescing bubbles of air in water[16][17][18] and is valid for cavities in materials, as well as particles in materials, as long as the surrounding material is essentially non-absorbing.

### Parasitology

It has also been used to study the structure of Plasmodium falciparum, a particularly pathogenic form of malaria.[19]

## Extensions

In 1986 P. A. Bobbert and J. Vlieger extended the Mie model to calculate scattering by a sphere in a homogeneous medium placed on flat surface. Like Mie model, the extended model can be applied to spheres with a radius close to the wavelength of the incident light.[20] There is a C++ code implementing Bobbert–Vlieger (BV) model.[21]

## References

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14. He, L; Kear-Padilla, L. L.; Lieberman, S. H.; Andrews, J. M. (2003). "Rapid in situ determination of total oil concentration in water using ultraviolet fluorescence and light scattering coupled with artificial neural networks". Analytica Chimica Acta. 478 (2): 245. doi:10.1016/S0003-2670(02)01471-X.
15. Lindner, H; Fritz, Gerhard; Glatter, Otto (2001). "Measurements on Concentrated Oil in Water Emulsions Using Static Light Scattering". Journal of Colloid and Interface Science. 242 (1): 239. Bibcode:2001JCIS..242..239L. doi:10.1006/jcis.2001.7754.
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17. Lentz, W. J.; Atchley, Anthony A.; Gaitan, D. Felipe (May 1995). "Mie scattering from a sonoluminescing air bubble in water". Applied Optics. 34 (15): 2648–54. Bibcode:1995ApOpt..34.2648L. doi:10.1364/AO.34.002648. PMID 21052406.
18. Gompf, B.; Pecha, R. (May 2000). "Mie scattering from a sonoluminescing bubble with high spatial and temporal resolution". Physical Review E. 61 (5): 5253–5256. Bibcode:2000PhRvE..61.5253G. doi:10.1103/PhysRevE.61.5253.
19. Serebrennikova, Yulia M.; Patel, Janus; Garcia-Rubio, Luis H. (2010). "Interpretation of the ultraviolet-visible spectra of malaria parasite Plasmodium falciparum". Applied Optics. 49 (2): 180–8. Bibcode:2010ApOpt..49..180S. doi:10.1364/AO.49.000180. PMID 20062504.
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