Definitions of polarization: depending on the time behavior of the electric field at some measurement point, it may be linear, circular, or a combination (elliptical). In general a beam will be partially polarized. A complete description of the polarization state can be found in the Stokes parameters. Note that, for example, Jackson uses senses of circular polarization opposite to what astronomers (and electrical engineers) usually consider. We use the astronomical convention: the direction of rotation of the E field follows a right-hand rule with respect to the propagation direction for pure right circular polarization. It is conventional to refer to polarization planes by the direction of the electric vector. There are multiple ways to decompose polarized light; for example, an elliptically polarized beam may be considered as the superposition of two oppositely circularly polarized beams of different intensity, equal frequency, and some initial phase shift in the desired reference frame.
Polarizing components: polarizers, depolarizers in optical domain. Birefringence, reflection. Use of, for example, rotating half-wave plates or Pockels cells (electrically controlled birefringence) to chop between measurement of the linearly polarized and unpolarized components. Polarimetrists are allergic to nonconcentric optical systems, since any non-normal reflection produces instrumental polarization, which can easily swamp weak polarization signals of interest.
Spectropolarimetry: one may introduce polarization elements into a spectrograph and then measure polarization properties as functions of wavelength. Chopping and simultaneous sky measurement are crucial here; only in special cases (i.e. above the atmosphere) can one simply take successive spectra through a rotating polarizer.
Imaging polarimetry with filters and (sliced images) with calcite splitters
Night-sky polarization, especially from scattered city lights
Rapid chopping and dual beams
Sources of instrumental signature: the instrument may be less than perfectly efficient, mixing slightly between polarized and unpolarized radiation. It may also introduce its own rotation into the plane of polarization (most astronomical applications are for linear polarization). These must be dealt with by observation of suitable reference stars; highly polarized bright stars are not common! Sometimes either the instrument or the whole telescope can rotate about the optical axis to guard against these effects; instrumental polarization will not be locked to celestial coordinates, so a long observation on an altazimuth mount gives much the same effect, and thus is used in radio astronomy to untangle instrumental and astronomical polarization.
Error analysis: beware simply transforming to the usual P, θ at low signal-to-noise, since we really measure some function of the Stokes parameters. There will be a bias toward higher polarization if one does, since polarization is always nonnegative. This is properly done in the Stokes domain.
Polarization in the X-ray and radio can also be measured. Radio polarimetry is particularly rich, since Faraday rotation and depolarization tell us about intervening magnetic fields (which are like ghosts in astronomy - even if you don't believe in them, you're still afraid of them.)
Why bother? Some physical mechanisms that give polarization are synchrotron emission, scattering by preferentially aligned dust grains, scattering in a nonaxisymmetric environment (like a flattened star's atmosphere), and scattering of hidden sources (like some Sy 2 nuclei and Herbig-Haro objects). There is also Faraday rotation and depolarization. In strongly magnetic stars, Zeeman splitting changes with polarization state.
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