Polarization mixing

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Concept in optics

title: "Polarization mixing" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["polarization-(waves)", "radiometry"] description: "Concept in optics" topic_path: "general/polarization-waves" source: "https://en.wikipedia.org/wiki/Polarization_mixing" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Concept in optics ::

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.

Example: A sloping, specular surface

::figure[src="https://upload.wikimedia.org/wikipedia/commons/f/f5/U_geo.png" caption="Geometry of a polarimetric detector relative to a sloping surface."] ::

The definition of the four Stokes components are, in a fixed basis:

: \left[ \begin{array}{c} I \ Q \ U \ V \end{array} \right] = \left[ \begin{array}{c} |E_v|^2 + |E_h|^2 \ |E_v|^2 - |E_h|^2 \ 2 \operatorname{Re}\left\langle E_v E_h^* \right\rangle \ 2 \operatorname{Im}\left\langle E_v E_h^* \right\rangle \end{array} \right],

where Ev and Eh are the electric field components in the vertical and horizontal directions respectively. The definitions of the coordinate bases are arbitrary and depend on the orientation of the instrument. In the case of the Fresnel equations, the bases are defined in terms of the surface, with the horizontal being parallel to the surface and the vertical in a plane perpendicular to the surface.

When the bases are rotated by 45 degrees around the viewing axis, the definition of the third Stokes component becomes equivalent to that of the second, that is the difference in field intensity between the horizontal and vertical polarizations. Thus, if the instrument is rotated out of plane from the surface upon which it is looking, this will give rise to a signal. The geometry is illustrated in the above figure: \theta is the instrument viewing angle with respect to nadir, \theta_\mathrm{eff} is the viewing angle with respect to the surface normal and \alpha is the angle between the polarisation axes defined by the instrument and that defined by the Fresnel equations, i.e., the surface.

Ideally, in a polarimetric radiometer, especially a satellite mounted one, the polarisation axes are aligned with the Earth's surface, therefore we define the instrument viewing direction using the following vector:

:\mathbf{\hat{v}} = (\sin \theta, ~0, ~\cos \theta).

We define the slope of the surface in terms of the normal vector, \mathbf{\hat{n}}, which can be calculated in a number of ways. Using angular slope and azimuth, it becomes:

:\mathbf{\hat{n}}=(\cos \psi \sin \mu,~~\sin \psi \cos \mu,~~\cos \mu),

where \mu is the slope and \psi is the azimuth relative to the instrument view. The effective viewing angle can be calculated via a dot product between the two vectors:

:\theta_\mathrm{eff} = \cos^{-1}(\mathbf{\hat{n}} \cdot \mathbf{\hat{v}}),

from which we compute the reflection coefficients, while the angle of the polarisation plane can be calculated with cross products:

: \alpha = \mathrm{sgn}(\mathbf{\hat{n}} \cdot \mathbf{\hat{j}}) \cos^{-1}\left( \frac { \mathbf{\hat{j}} \cdot \mathbf{\hat{n}} \times \mathbf{\hat{v}} } { |\mathbf{\hat{n}} \times \mathbf{\hat{v}}| } \right),

where \mathbf{\hat{j}} is the unit vector defining the y-axis.

The angle, \alpha, defines the rotation of the polarization axes between those defined for the Fresnel equations versus those of the detector. It can be used to correct for polarization mixing caused by a rotated detector, or to predict what the detector "sees", especially in the third Stokes component. See Stokes parameters#Relation to the polarization ellipse.

Application: Aircraft radiometry data

Correcting or removing bad data

Predicting U

| width = 300 | footer = | image1 = U dep.png | alt1 = | caption1 = Dependence of eU on surface-slope and azimuth-angle for a refractive index of 2 and a nominal instrument pointing-angle of 45 degrees. | image2 = Ugotit.png | alt2 = | caption2 = Modelled U versus Pol-Ice field data for a circular overflight over sea ice. Many of the radiance measurements over sea ice included large signals in the third Stoke component, U. It turns out that these can be predicted to fairly high accuracy simply from the aircraft attitude. We use the following model for emissivity in U:

:e_U = \sqrt{e_v^2 - e_h^2} \sin (2 \alpha)

where eh and ev are the emissivities calculated via the Fresnel or similar equations and eU is the emissivity in U—that is, U = e_U T, where T is physical temperature—for the rotated polarization axes. The plot below shows the dependence on surface-slope and azimuth angle for a refractive index of 2 (a common value for sea ice | author=M. R. Vant | author2=R. O. Ramseier | author3=V. Makios | name-list-style=amp | journal=Journal of Applied Physics | date=1978 | title=The complex-dielectric constant of sea ice at frequencies in the range 0.1-4.0 GHz | volume=49 | pages=1246–1280 | doi = 10.1063/1.325018 | issue=3 | bibcode=1978JAP....49.1264V

References

::callout[type=info title="Wikipedia Source"] This article was imported from Wikipedia and is available under the Creative Commons Attribution-ShareAlike 4.0 License. Content has been adapted to SurfDoc format. Original contributors can be found on the article history page. ::