Polarization

Characterization of Polarization

(Chat Hull, edited by Kara Kundert)

Introduction to Polarization

Definition

A source is considered partially polarized if any polarization is more common than its orthogonal state. In other words, if there is any directional preference in the electric field of a source, then it is polarized.

Causes

Polarization frequently arises because of magnetic fields. We might desire to detect polarized emission to better understand the following:

• Synchrotron radiation – from electrons spinning around magnetic field lines
• Pulsar radiation – highly collimated beams of radiation from the magnetic poles of rapidly rotating neutron stars
• Faraday rotation – position angles of linear polarization rotate as the light propagates through a region with line-of-sight magnetic field
• Zeeman splitting – spectral lines are split into multiple lines due to the presence of a static magnetic field
• Dust emission – dust grains orient themselves perpendicular to the ambient magnetic field

Benefits of measurement

When you measure a single polarization, you only get half of the radiation (assuming an unpolarized source)! So when you measure both polarizations, you get a factor of ${\displaystyle {\sqrt {2}}}$ in sensitivity, in addition to the ability to investigate the scientific questions answerable with full-Stokes observations.

Polarization ellipse

Say you’re observing a particle spinning non-relativistically around a magnetic field. If you view the system edge-on (i.e. in the plane of the rotation), you observe linear polarization. If you observe the system pole-on, you see circular polarization.

But what about everywhere in between? This is where you observe the generic polarization ellipse:

The polarization ellipse.
• Note the position angle ${\displaystyle \Psi }$
• Linear polarization: when the orthogonal vibrations have the same phase. Amplitudes can be different or the same.
• Circular polarization: when the orthogonal vibrations have exactly a 90${\displaystyle ^{\circ }}$ phase difference and have the same amplitude.
• Elliptical polarization: when the amplitudes of the orthogonal vibrations are different and there’s any phase difference; or when the amplitudes are the same and there’s a phase difference ${\displaystyle \phi \neq 90^{\circ }}$.

Views of the polarization ellipse as a function of ${\displaystyle E_{x}/E_{y}}$ as well as phase angle between the two E-field vectors.

Position angle of polarization

The position angle ${\displaystyle \Psi }$ is the angle of orientation of the ellipse relative to the two chosen orthogonal bases (say, the vertical and horizontal axes of your telecope’s field of view). If you observed linear polarization, the angle of the linear polarization would also be ${\displaystyle \Psi }$.

${\displaystyle \Psi }$ is periodic in ${\displaystyle \pi }$, not in 2${\displaystyle \pi }$, which means that it has an orientation, but not a direction! The "iron filings" on polarization maps only show the orientation of the magnetic field, not its direction.

Definition of handedness (LCP vs. RCP)

The IEEE (Institute of Electrical and Electronics Engineers) has defined handedness as pictured above in the second figure. Light is right-circularly polarized (RCP) when the radiation phasor travels around to the right as viewed by the source. As we look up in the sky, RCP looks as if it’s going around to the left.

Stokes parameters

Stokes parameters are linear combinations of power measured in orthogonal polarizations. By using four Stokes parameters, you can fully describe a polarized wave having four parameters:

• ${\displaystyle X}$-amplitude
• ${\displaystyle Y}$-amplitude
• Phase angle ${\displaystyle \phi }$ between them
• Polarization fraction ${\displaystyle \Pi }$

Stokes I: total intensity

Stokes ${\displaystyle I}$ represents the total power in the incoming radiation. You can get the total power by sampling two orthogonal polarizations (no matter which two they are).

${\displaystyle I=E_{0^{\circ }}^{2}+E_{90^{\circ }}^{2}\,\!}$

Stokes Q & U: linear polarization

{\displaystyle {\begin{aligned}Q&=E_{0^{\circ }}^{2}-E_{90^{\circ }}^{2}\\U&=E_{45^{\circ }}^{2}-E_{-45^{\circ }}^{2}\end{aligned}}\,\!}

${\displaystyle Q}$ and ${\displaystyle U}$ fully describe linear polarization. If, for example, you only had ${\displaystyle Q}$, then ${\displaystyle Q=0}$ for light polarized at 45${\displaystyle ^{\circ }}$! Not good.

More simply, you can say that for linear polarization, you need to find both an amplitude and a position angle. To solve for both parameters, you need two equations.

${\displaystyle Q}$ and ${\displaystyle U}$ can also be written this way:

{\displaystyle {\begin{aligned}{\frac {Q}{I}}&=p_{QU}\cos \left(2\Psi \right)\\{\frac {U}{I}}&=p_{QU}\sin \left(2\Psi \right)\\\end{aligned}}\,\!}

where ${\displaystyle p_{QU}}$ is the total fractional linear polarization:

${\displaystyle p_{QU}={\frac {\sqrt {Q^{2}+U^{2}}}{I}}\,\,.\,\!}$

The argument of the above sin and cos is ${\displaystyle 2\Psi }$ because ${\displaystyle \Psi }$ is periodic in ${\displaystyle \pi }$.

You can also take the above two equations and divide them to get a relation between ${\displaystyle Q}$, ${\displaystyle U}$, and ${\displaystyle \Psi }$:

${\displaystyle \Psi =0.5\arctan {\frac {U}{Q}}\,\!}$

Stokes V: circular polarization

Stokes ${\displaystyle V}$ deals with circular polarization, and is defined as follows:

${\displaystyle V=E_{RCP}^{2}-E_{LCP}^{2}\,\,.\,\!}$

I.e., a net RCP signal has a positive Stokes V. Note: in 1974 the IAU established this convention; however, many publications define Stokes V as LCP-RCP! Be sure to check the definition in previously published work, and to report which convention you’ve used in your work.

The fractional circular polarization is:

${\displaystyle p_{V}={\frac {V}{I}}\,\,.\,\!}$

Polarization fraction

The total fraction of polarized light is calculated in the following way:

{\displaystyle {\begin{aligned}\Pi &={\frac {\sqrt {Q^{2}+U^{2}+V^{2}}}{I}}\\\Pi &\leq 1\end{aligned}}\,\!}

If both ${\displaystyle p_{QU}}$ and ${\displaystyle p_{V}}$ are nonzero, the polarization is elliptical.

Remember that you can get total intensity by adding the intensities of two orthogonal polarizations. So, if you were to take any of the differences listed below and turn them into sums, you’d get total intensity. For example:

{\displaystyle {\begin{aligned}I&=E_{0^{\circ }}^{2}+E_{90^{\circ }}^{2}\\&=E_{45^{\circ }}^{2}+E_{-45^{\circ }}^{2}\\&=E_{LCP}^{2}+E_{RCP}^{2}\end{aligned}}\,\!}

A note about imaging Stokes parameters

Stokes parameters are actual intensities! This means that once you’ve done your data reduction, you can actually map all four parameters on your image.

To combine polarizations, you must convert to Stokes parameters, average those, and then convert back to intensity. Averaging intensities (which are always positive) or angles will result in wacky (generally incorrect) answers.

A pair of left and right circularly polarized receivers can be used to equally match linear polarizations. However, reception of circularly polarized signals on linear feeds entails a 3dB reduction in the signal-to-noise, as the received power has necessarily been cut in half.

Measurement of Polarization

(Nicholas McConnell)

Full-Stokes observations

Your dual-polarization receivers generally comprise either crossed linear or crossed circular feeds. Using the voltages detected by those orthogonal feeds, you can get all four Stokes parameters by calculating the following products:

{\displaystyle {\begin{aligned}I&=E_{X}E_{X}^{*}+E_{Y}E_{Y}^{*}\\Q&=E_{X}E_{X}^{*}-E_{Y}E_{Y}^{*}\\U&=E_{X}E_{Y}^{*}+E_{X}^{*}E_{Y}\\iV&=E_{X}E_{Y}^{*}-E_{X}^{*}E_{Y}\end{aligned}}\,\!}

where ${\displaystyle E_{X}^{*}}$ denotes the complex conjugate of ${\displaystyle E_{X}}$. You measure the complex conjugate by introducing a 90${\displaystyle ^{\circ }}$ phase delay. A figure depicting the products follows: \

Voltage multiplication.

We now continue to describe two matrices that are useful for describing distortions in a polarized signal: Jones and Mueller matrices.

Jones matrix

A Jones matrix describes the translation between the intrinsic and measured electric field.

${\displaystyle \left[{\begin{array}{cc}E_{X}\\E_{Y}\end{array}}\right]_{output}={\textbf {J}}\left[{\begin{array}{cc}E_{X}\\E_{Y}\end{array}}\right]_{intrinsic}\,\!}$

where J is the Jones matrix.

Mueller matrix

The Mueller matrix is similar to the Jones matrix, but instead of describing the electric field, it relates the intrinsic and measured Stokes parameters.

${\displaystyle \left[{\begin{array}{cc}I\\Q\\U\\V\end{array}}\right]_{output}={\textbf {M}}\left[{\begin{array}{cc}I\\Q\\U\\V\end{array}}\right]_{intrinsic}\,\!}$

where ${\displaystyle {\textbf {M}}}$ is the Mueller matrix. Ultimately, you get the intrinsic signal by measuring ${\displaystyle {\textbf {M}}}$ and then inverting it to get the intrinsic Stokes parameters.

Contributions to the Mueller matrix

The distortion experienced by a polarized signal comes from a variety of places, both hardware- and sky-related.

${\displaystyle {\textbf {M}}\sim {\textbf {M}}_{electronics}\,{\textbf {M}}_{feed}\,{\textbf {M}}_{sky}\,\!}$
• ${\displaystyle {\textbf {M}}_{electronics}}$: Gain calibration (${\displaystyle \Delta G}$) – ${\displaystyle E_{X}}$ and ${\displaystyle E_{Y}}$ have different signal paths, and potentially different gains. This gives you false linear polarization.
• ${\displaystyle {\textbf {M}}_{electronics}}$: Phase calibration (${\displaystyle \Delta \Psi }$) – correlated noise source has different phase in different signal paths. This gives you false circular polarization.
• ${\displaystyle {\textbf {M}}_{feed}}$: "Leakage" – some of each polarization "leaks" into the orthogonal polarization. Causes: non-orthogonal dipoles or feed substructure; cross-talk between dipoles; imperfect phase shift (e.g. between RCP and LCP); frequency-dependent axis wobble.
• ${\displaystyle {\textbf {M}}_{sky}}$: Position angle rotation and Faraday rotation from the ionosphere.

Calibration to find Mueller matrix

Observe one source at many different hour angles; preferably a source with high ${\displaystyle Q}$ and ${\displaystyle U}$, but low ${\displaystyle V}$, for linear feeds. For circular feeds the calibration source should have some linear and circular polarization.

By calibrating this way, you’ll note that instrument-caused polarization doesn’t rotate (relative to the orthogonal feeds) as a function of hour angle. When calibrating, it’s best to observe the source near zenith, where the position angle of its polarization is changing the fastest.

Polarization structure of the telescope beam

If a source is not perfectly isotropic and on-axis, then additional, beam-based distortions are introduced.

Beam squash

The power responses of the linearly polarized feeds are oriented perpendicular to one another. When you subtract the responses to measure ${\displaystyle Q}$, you get a clover-leaf pattern. \

Beam squash.

Beam squint

If the feeds of the telescope are not perfectly aligned with the focal point of the primary reflector, opposite circular polarizations will be most strongly detected at different locations on the sky. When you subtract the responses to measure ${\displaystyle V}$, you get a double-lobed pattern. \

Beam squint.

Off-axis source detection

When you observe an off-axis source, the component of the electric field parallel to the dish surface will be preserved, and the component perpendicular to the surface will be reduced, introducing false polarization.

Other depolarization effects

Bandwidth depolarization

As a polarized ray propagates through a region with a line-of-sight magnetic field component, the position angle ${\displaystyle \phi }$ of the radiation shifts:

${\displaystyle \Delta \phi \propto {\frac {\int {n_{e}B_{||}}}{\nu ^{2}}}\,\!}$

This is known as Faraday rotation. At lower frequencies, the phase can wrap so quickly that the light can effectively become depolarized if the frequency resolution of the telescope isn’t high enough to detect the rotation of the position angle between each frequency channel.

Of course, Faraday rotation can be both a bother (ionosphere) or a desired quantity (for probing interstellar fields). To remove the component of Faraday rotation from the ionosphere, one must first model the Earth’s magnetic field and then model the electron density in the ionosphere.

Bandwidth depolarization.

Optical depth depolarization

This happens when a plasma is optically thin, and you’re rays originating at various depths, all of which experience different amounts of rotation before exiting the plasma. \

Optical depth depolarization.

Beam depolarization

This occurs when the beam of the telescope is large enough to encompass several sources with different polarization angles, all of which average out across the beam. A possible way around this is rotation-measure (RM) synthesis (discussed in Casey Law’s recent work), which does a Fourier-transform-type analysis of the incoming radiation to pick off peaks of different RM.

Beam depolarization.