In electromagnetism, 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 German physicist 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, the "Mie scattering" formulas are most useful in 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 scattering or aerosol particle scattering) takes place in the lower 4,500 m (15,000 ft) of the atmosphere, where many essentially spherical particles with diameters approximately equal to the wavelength of the incident ray may be present. 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 spherical harmonics. The internal field is expanded into regular vector spherical harmonics. 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 behavior of the system. But for objects whose size is within a few orders of magnitude of 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 makes it a particularly useful formalism when using scattered light to measure particle size.

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, λ is the wavelength of light under consideration, 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 that of the environment, and its size is much smaller in comparison to the wavelength of light divided by |n − 1|, where n is the refractive index:^[3]

${\displaystyle {\begin{aligned}|n-1|&\ll 1\\kd|n-1|&\ll 1\end{aligned}}}$

where ${\textstyle k}$ is the wavevector of the light (${\textstyle k={\frac {2\pi }{\lambda }}}$), and ${\displaystyle d}$ refers to the linear dimension of the particle. The former condition is often referred as optically soft and the approximation holds for particles of arbitrary shape.^[3]

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, 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

The scattering by a spherical nanoparticle is solved exactly regardless of the particle size. We consider scattering by a plane wave propagating along the z-axis polarized along the x-axis. Dielectric and magnetic permeabilities of a particle are ${\displaystyle \varepsilon _{1}}$ and ${\displaystyle \mu _{1}}$, and ${\displaystyle \varepsilon }$ and ${\displaystyle \mu }$ for the environment.

In order to solve the scattering problem,^[3] we write first the solutions of the vector Helmholtz equation in spherical coordinates, since the fields inside and outside the particles must satisfy it. Helmholtz equation:

${\displaystyle \nabla ^{2}\mathbf {E} +{k}^{2}\mathbf {E} =0,\quad \nabla ^{2}\mathbf {H} +{k}^{2}\mathbf {H} =0.}$

In addition to the Helmholtz equation, the fields must satisfy the conditions ${\displaystyle \nabla \cdot \mathbf {E} =\nabla \cdot \mathbf {H} =0}$ and ${\displaystyle \nabla \times \mathbf {E} =i\omega \mu \mathbf {H} }$, ${\displaystyle \nabla \times \mathbf {H} =-i\omega \varepsilon \mathbf {E} }$. Vector spherical harmonics possess all the necessary properties, introduced as follows:

${\displaystyle \mathbf {M} _{^{e}_{o}mn}=\nabla \times \left(\mathbf {r} \psi _{^{e}_{o}mn}\right)}$ — magnetic harmonics (TE),

${\displaystyle {\mathbf{N}}_{{}_o^{e}mn}=\frac{\mathrm{\nabla <!--\; \nabla \; -->}\times <!--\; \times \; -->{\mathbf{M}}_{{}_o^{e}mn}}{}}$Zdroj:https://en.wikipedia.org?pojem=Mie_scattering
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