Vector radiative transfer

Last updated April 08, 2019

In spectroscopy and radiometry, vector radiative transfer (VRT) is a method of modelling the propagation of polarized electromagnetic radiation in low density media. In contrast to scalar radiative transfer (RT), which models only the first Stokes component, the intensity, VRT models all four components through vector methods.

Spectroscopy is the study of the interaction between matter and electromagnetic radiation. Historically, spectroscopy originated through the study of visible light dispersed according to its wavelength, by a prism. Later the concept was expanded greatly to include any interaction with radiative energy as a function of its wavelength or frequency, predominantly in the electromagnetic spectrum, though matter waves and acoustic waves can also be considered forms of radiative energy; recently, with tremendous difficulty, even gravitational waves have been associated with a spectral signature in the context of LIGO and laser interferometry. Spectroscopic data are often represented by an emission spectrum, a plot of the response of interest as a function of wavelength or frequency.

Radiometry is a set of techniques for measuring electromagnetic radiation, including visible light. Radiometric techniques in optics characterize the distribution of the radiation's power in space, as opposed to photometric techniques, which characterize the light's interaction with the human eye. Radiometry is distinct from quantum techniques such as photon counting.

In physics, electromagnetic radiation refers to the waves of the electromagnetic field, propagating (radiating) through space, carrying electromagnetic radiant energy. It includes radio waves, microwaves, infrared, (visible) light, ultraviolet, X-rays, and gamma rays.

For a single frequency, $\nu$ , the VRT equation for a scattering media can be written as follows:

{\frac {\mathrm {d} {\vec {I}}({\hat {n}},\nu )}{\mathrm {d} s}}=-\mathbf {K} {\vec {I}}+{\vec {a}}B(\nu ,T)+\int _{4\pi }\mathbf {Z} ({\hat {n}},{\hat {n}}^{\prime },\nu ){\vec {I}}\mathrm {d} {\hat {n}}^{\prime }

where s is the path, ${\hat {n}}$ is the propagation vector, K is the extinction matrix, ${\vec {a}}$ is the absorption vector, B is the Planck function and Z is the scattering phase matrix.

All the coefficient matrices, K, ${\vec {a}}$ and Z, will vary depending on the density of absorbers/scatterers present and must be calculated from their density-independent quantities, that is the attenuation coefficient vector, ${\vec {a}}$ , is calculated from the mass absorption coefficient vector times the density of the absorber. Moreover, it is typical for media to have multiple species causing extinction, absorption and scattering, thus these coefficient matrices must be summed up over all the different species.

The attenuation coefficient or narrow-beam attenuation coefficient characterizes how easily a volume of material can be penetrated by a beam of light, sound, particles, or other energy or matter. A large attenuation coefficient means that the beam is quickly "attenuated" (weakened) as it passes through the medium, and a small attenuation coefficient means that the medium is relatively transparent to the beam. The SI unit of attenuation coefficient is the reciprocal metre (m⁻¹). Extinction coefficient is an old term for this quantity but is still used in meteorology and climatology. Most commonly, the quantity measures the number of downward e-foldings of the original intensity that will be had as the energy passes through a unit thickness of material, so that an attenuation coefficient of 1 m^-1 means that after passing through 1 metre, the radiation will be reduced by a factor of e, and for material with a coefficient of 2 m^-1, it will be reduced twice by e, or e². Other measures may use a different factor than e, such as the decadic attenuation coefficient below.

Extinction is caused both by simple absorption as well as from scattering out of the line-of-sight, ${\hat {n}}$ , therefore we calculate the extinction matrix from the combination of the absorption vector and the scattering phase matrix:

In physics, absorption of electromagnetic radiation is how matter takes up a photon's energy — and so transforms electromagnetic energy into internal energy of the absorber. A notable effect (attenuation) is to gradually reduce the intensity of light waves as they propagate through a medium. Although the absorption of waves does not usually depend on their intensity, in certain conditions (optics) the medium's transparency changes by a factor that varies as a function of wave intensity, and saturable absorption occurs.

\mathbf {K} ({\hat {n}},\nu )={\vec {a}}(\nu )\mathbf {I} +\int _{4\pi }\mathbf {Z} ({\hat {n}}^{\prime },{\hat {n}},\nu )\mathrm {d} {\hat {n}}^{\prime }

where I is the identity matrix.

The four-component radiation vector, ${\vec {I}}=(I,Q,U,V)$ where I, Q, U and V are the first through fourth elements of the Stokes parameters, respectively, fully describes the polarization state of the electromagnetic radiation. It is this vector-nature that considerably complicates the equation. Absorption will be different for each of the four components, moreover, whenever the radiation is scattered, there can be a complex transfer between the different Stokes components—see polarization mixing—thus the scattering phase function has 4*4=16 components. It is, in fact, a rank-two tensor.

The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation. They were defined by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse. The effect of an optical system on the polarization of light can be determined by constructing the Stokes vector for the input light and applying Mueller calculus, to obtain the Stokes vector of the light leaving the system. The parameters were named after Stokes first by Subrahamanyan Chandrasekhar.

In optics, polarization mixing refers to changes in the relative strengths of the Stokes parameters caused by reflection or scattering—see vector radiative transfer—or by changes in the radial orientation of the detector.

In mathematics, a tensor is a geometric object that maps in a multi-linear manner geometric vectors, scalars, and other tensors to a resulting tensor. Vectors and scalars which are often used in elementary physics and engineering applications, are considered as the simplest tensors. Vectors from the dual space of the vector space, which supplies the geometric vectors, are also included as tensors. Geometric in this context is chiefly meant to emphasize independence of any selection of a coordinate system.

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References

Claudia Emde (2004). A polarized discrete ordinate scattering model for radiative transfer simulations in spherical atmospheres with thermal source (Thesis). University of Bremen. External link in |title= (help)

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