Polarization mixing
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In
optics Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...
, polarization mixing refers to changes in the relative strengths of the
Stokes parameters 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 incoher ...
caused by
reflection Reflection or reflexion may refer to: Science and technology * Reflection (physics), a common wave phenomenon ** Specular reflection, reflection from a smooth surface *** Mirror image, a reflection in a mirror or in water ** Signal reflection, in ...
or
scattering Scattering is a term used in physics to describe a wide range of physical processes where moving particles or radiation of some form, such as light or sound, are forced to deviate from a straight trajectory by localized non-uniformities (including ...
—see
vector radiative transfer 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 ...
—or by changes in the radial orientation of the detector.


Example: A sloping, specular surface

The definition of the four Stokes components are, in a fixed
basis Basis may refer to: Finance and accounting *Adjusted basis, the net cost of an asset after adjusting for various tax-related items *Basis point, 0.01%, often used in the context of interest rates *Basis trading, a trading strategy consisting of ...
: : \left \begin I \\ Q \\ U \\ V \end \right= \left E_v, ^2 + , E_h, ^2 \\ , E_v, ^2 - , E_h, ^2 \\ 2 \operatorname\left\langle E_v E_h^* \right\rangle \\ 2 \operatorname\left\langle E_v E_h^* \right\rangle \end \right where ''E''v and ''E''h are the
electric field An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
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 Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
, the bases are defined in terms of the surface, with the horizontal being parallel to the surface and the vertical in a plane
perpendicular In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
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 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 Polarimetry is the measurement and interpretation of the polarization of transverse waves, most notably electromagnetic waves, such as radio or light waves. Typically polarimetry is done on electromagnetic waves that have traveled through or ha ...
radiometer A radiometer or roentgenometer is a device for measuring the radiant flux (power) of electromagnetic radiation. Generally, a radiometer is an infrared radiation detector or an ultraviolet detector. Microwave radiometers operate in the microwave w ...
, 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 = (\sin \theta, ~0, ~\cos \theta). We define the slope of the surface in terms of the normal vector, \mathbf, which can be calculated in a number of ways. Using angular slope and azimuth, it becomes: :\mathbf=(\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 In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
between the two vectors: :\theta_\mathrm = \cos^(\mathbf \cdot \mathbf), from which we compute the reflection coefficients, while the angle of the polarisation plane can be calculated with cross products: : \alpha = \mathrm(\mathbf \cdot \mathbf) \cos^\left( \frac \right), where \mathbf 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

The Pol-Ice 2007 campaign included measurements over
sea ice Sea ice arises as seawater freezes. Because ice is less dense than water, it floats on the ocean's surface (as does fresh water ice, which has an even lower density). Sea ice covers about 7% of the Earth's surface and about 12% of the world's oce ...
and open water from a fully polarimetric, aeroplane-mounted, L-band (1.4 GHz)
radiometer A radiometer or roentgenometer is a device for measuring the radiant flux (power) of electromagnetic radiation. Generally, a radiometer is an infrared radiation detector or an ultraviolet detector. Microwave radiometers operate in the microwave w ...
. Since the radiometer was fixed to the aircraft, changes in
aircraft attitude Flight dynamics is the science of air vehicle orientation and control in three dimensions. The three critical flight dynamics parameters are the angles of rotation in three dimensions about the vehicle's center of gravity (cg), known as ''pitc ...
are equivalent to changes in surface slope. Moreover,
emissivity The emissivity of the surface of a material is its effectiveness in emitting energy as thermal radiation. Thermal radiation is electromagnetic radiation that most commonly includes both visible radiation (light) and infrared radiation, which is n ...
over calm water and to a lesser extent, sea ice, can be effectively modelled using the
Fresnel equations The Fresnel equations (or Fresnel coefficients) describe the reflection and transmission of light (or electromagnetic radiation in general) when incident on an interface between different optical media. They were deduced by Augustin-Jean Fresne ...
. Thus this is an excellent source of data with which to test the ideas discussed in the previous section. In particular, the campaign included both circular and
zig-zag A zigzag is a pattern made up of small corners at variable angles, though constant within the zigzag, tracing a path between two parallel lines; it can be described as both jagged and fairly regular. In geometry, this pattern is described as a ...
ging overflights which will produce strong mixing in the Stokes parameters.


Correcting or removing bad data

To test the calibration of the EMIRAD II radiometer used in the Pol-Ice campaign, measurements over open water were compared with model results based on the Fresnel equations. The first plot, which compares the measured data with the model, shows that vertically polarized channel is too high, but more importantly in this context, are the smeared points in between the otherwise relatively clean function for measured vertical and horizontal
brightness temperature Brightness temperature or radiance temperature is the temperature at which a black body in thermal equilibrium with its surroundings would have to be in order to duplicate the observed intensity of a grey body object at a frequency \nu. This conce ...
as a function of
viewing angle In display technology parlance, viewing angle is the angle at which a display can be viewed with acceptable visual performance. In a technical context, the angular range is called viewing cone defined by a multitude of viewing directions. The vie ...
. These are the result of polarization mixing caused by changes in the attitude of the aircraft, particularly the
roll angle In physics, the degrees of freedom (DOF) of a mechanical system is the number of independent parameters that define its configuration or state. It is important in the analysis of systems of bodies in mechanical engineering, structural engineering ...
. Since there are plenty of data points, rather than correcting the bad data, the authors simply exclude points for which the angle, \alpha, is too large. The result is shown at far right.


Predicting U

Many of the
radiance In radiometry, radiance is the radiant flux emitted, reflected, transmitted or received by a given surface, per unit solid angle per unit projected area. Radiance is used to characterize diffuse emission and reflection of electromagnetic radiatio ...
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 \sin (2 \alpha) where ''e''h and ''e''v are the emissivities calculated via the Fresnel or similar equations and ''e''U 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 An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north. Mathematicall ...
angle for a
refractive index In optics, the refractive index (or refraction index) of an optical medium is a dimensionless number that gives the indication of the light bending ability of that medium. The refractive index determines how much the path of light is bent, or ...
of 2 (a common value for sea ice ) and a nominal instrument pointing-angle of 45 degrees. Using the same model, we can simulate the ''U''-component of the Stokes vector for the radiometer.


See also

*
Polarization scrambling Polarization scrambling is the process of rapidly varying the Polarization (waves), polarization of light within a system using a polarization controller so that the average polarization over time is effectively randomized. Polarization scrambling c ...
*
Stokes parameters 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 incoher ...


References

{{reflist Polarization (waves) Radiometry