View source for Stokes parameters

To characterize the radiance (intensity) or flux and state of polarization of a beam of electromagnetic radiation (cf. also [[Electromagnetism|Electromagnetism]]) one can use four real parameters which have the same physical dimension. These so-called Stokes parameters were first introduced by G.C. Stokes [[#References|[a7]]] in 1852. It took about a hundred years before Stokes parameters were used on a large scale in optics and theories of light scattering by molecules and small particles. (See, e.g., [[#References|[a1]]], [[#References|[a2]]], [[#References|[a3]]], [[#References|[a5]]], [[#References|[a4]]], [[#References|[a6]]], [[#References|[a8]]].)

To define the Stokes parameters, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305801.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305802.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305803.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305804.png" />, one first considers a monochromatic beam of electromagnetic radiation. One defines two orthogonal unit vectors <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305806.png" /> such that the direction of propagation of the beam is the direction of the vector product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305807.png" />. The components of the electric field vectors at a point, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305808.png" />, in the beam can be written as

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s1305809.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058010.png" /> is the circular frequency, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058011.png" /> is time, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058012.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058013.png" /> are (non-negative) amplitudes. One now defines the Stokes parameters by

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058014.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058015.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a3)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058016.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a4)</td></tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a5)</td></tr></table>

The end point of the electric vector at a point, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058018.png" />, in the beam describes an ellipse, the so-called polarization ellipse, whose ellipticity and orientation with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058020.png" /> follow from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058023.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058024.png" />, the electric vector at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058025.png" /> moves clockwise, as viewed by an observer looking in the direction of propagation. Clearly, the following relation holds:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058026.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a6)</td></tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058027.png" /> for linearly polarized radiation and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058028.png" /> for circularly polarized radiation.

In general, electromagnetic waves are not exactly monochromatic, but the amplitudes <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058030.png" />, as well as the phase differences <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058031.png" />, may vary slowly in time. In this case the Stokes parameters are defined as before, with one exception, namely time averages must be taken on the right-hand sides of (a2)–(a5). The polarization may now be partial and the beam can be decomposed in a completely unpolarized and a completely polarized beam. The orientation and shape of the polarization ellipse of the latter beam is again given by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058032.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058033.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058034.png" />. The identity of (a6) is now replaced by the inequality

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058035.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a7)</td></tr></table>

and the ratio <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058036.png" /> is called the degree of polarization. For completely polarized radiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058037.png" />, for partially polarized radiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058038.png" />, and for unpolarized (natural) radiation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058039.png" />.

The Stokes parameters can be combined into a column vector with elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058042.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058043.png" />, called a Stokes vector. Stokes vectors of constituent beams are added to obtain the Stokes vector of a composite beam if no interference effects occur. Optical devices and processes like scattering and absorption can be described by real <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s130/s130580/s13058044.png" /> (Mueller) matrices that transform the Stokes vectors of primary beams into those of secondary beams.

The Stokes parameters as defined above are one of many possible representations of polarized radiation, several of which are only slight modifications of each other (see, e.g., [[#References|[a1]]]). Stokes parameters are also used in quantum mechanics in connection with polarization of elementary particles.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Born,   E. Wolf,   "Principles of optics. Electromagnetic theory of propagation, interference and diffraction of light" , Pergamon  (1993)  (Edition: Sixth)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Chandrasekhar,   "Radiative transfer" , Oxford Univ. Press  (1950)  (Reprint: Dover 1960)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.Z. Dolginov,   Yu.N. Gnedin,   N.A. Silant'ev,   "Propagation and polarization of radiation in cosmic media" , Gordon &amp; Breach  (1995)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  H.C. van de Hulst,   "Light scattering by small particles" , Wiley  (1957)  (Reprint: Dover 1981)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J.W. Hovenier,   C.V.M. van der Mee,   "Fundamental relationships relevant to the transfer of polarized light in a scattering atmosphere"  ''Astron. Astrophys.'' , '''128'''  (1983)  pp. 1–16</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  M.I. Mishchenko,   J.W. Hovenier,   L.D. Travis,   "Light scattering by nonspherical particles. Theory, measurements, and applications" , Acad. Press  (2000)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  G.C. Stokes,   "On the composition and resolution of streams of polarized light from different sources"  ''Trans. Cambridge Philos. Soc.'' , '''9'''  (1852)  pp. 399–416</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  F.T. Ulaby,   C. Elachi,   "Radar polarimetry for geoscience applications" , Artech House, Boston  (1990)</TD></TR></table>