Editing Stokes parameters (section)

==Alternative explanation==
[[File:Polarisation ellipse.svg|250px|right]]
A [[monochromatic]] [[plane wave]] is specified by its [[propagation vector]], <math>\vec{k}</math>, and the [[complex amplitude]]s of the [[electric field]], <math>E_1</math> and <math>E_2</math>, in a [[Basis (linear algebra)|basis]] <math>(\hat{\epsilon}_1,\hat{\epsilon}_2)</math>. The pair <math>(E_1, E_2)</math> is called a [[Jones vector]]. Alternatively, one may specify the propagation vector, the [[Phase (waves)|phase]], <math>\phi</math>, and the polarization state, <math>\Psi</math>, where <math>\Psi</math> is the curve traced out by the electric field as a function of time in a fixed plane. The most familiar polarization states are linear and circular, which are [[Degeneracy (mathematics)|degenerate]] cases of the most general state, an [[ellipse]].

One way to describe polarization is by giving the [[Semi-major axis|semi-major]] and [[Semi-minor axis|semi-minor]] axes of the polarization ellipse, its orientation, and the direction of rotation (See the above figure). The Stokes parameters <math>I</math>, <math>Q</math>, <math>U</math>, and <math>V</math>, provide an alternative description of the polarization state which is experimentally convenient because each parameter corresponds to a sum or difference of measurable intensities. The next figure shows examples of the Stokes parameters in degenerate states.

[[File:StokesParameters.png|center]]

===Definitions===
The Stokes parameters are defined by{{citation needed|date=December 2019}}

:<math> \begin{align}
I & \equiv \langle E_x^{2} \rangle + \langle E_y^{2} \rangle \\
 & = \langle E_a^{2} \rangle + \langle E_b^{2} \rangle \\
 & =  \langle E_r^{2} \rangle + \langle E_l^{2} \rangle, \\
Q & \equiv \langle E_x^{2} \rangle - \langle E_y^{2} \rangle, \\
U & \equiv \langle E_a^{2} \rangle - \langle E_b^{2} \rangle, \\
V & \equiv  \langle E_r^{2} \rangle - \langle E_l^{2} \rangle.
\end{align} </math>

where the subscripts refer to three different bases of the space of [[Jones vectors]]: the standard [[Cartesian coordinate system|Cartesian basis]] (<math>\hat{x},\hat{y}</math>), a Cartesian basis rotated by 45° (<math>\hat{a},\hat{b}</math>), and a circular basis (<math>\hat{l},\hat{r}</math>). The circular basis is defined so that <math>\hat{l} = (\hat{x}+i\hat{y})/\sqrt{2}</math>, <math>\hat{r} = (\hat{x}-i\hat{y})/\sqrt{2}</math>.

The symbols ⟨⋅⟩ represent [[expectation value]]s. The light can be viewed as a random variable taking values in the space ''C''<sup>2</sup> of [[Jones vectors]] <math>(E_1, E_2)</math>. Any given measurement yields a specific wave (with a specific phase, polarization ellipse, and magnitude), but it keeps flickering and wobbling between different outcomes. The expectation values are various averages of these outcomes. Intense, but unpolarized light will have ''I'' > 0 but 
''Q'' = ''U'' = ''V'' = 0, reflecting that no polarization type predominates. A convincing waveform is depicted at the article on [[Coherence (physics)#Measurement of spectral coherence|coherence]].

The opposite would be perfectly polarized light which, in addition, has a fixed, nonvarying amplitude—a pure sine curve. This is represented by a random variable with only a single possible value, say <math>(E_1, E_2)</math>. In this case one may replace the brackets by absolute value bars, obtaining a well-defined quadratic map{{citation needed|date=December 2019}}

:<math> \begin{matrix}
I \equiv |E_x|^{2} + |E_y|^{2} = |E_a|^{2} + |E_b|^{2} = |E_r|^{2} + |E_l|^{2} \\
Q \equiv |E_x|^{2} - |E_y|^{2}, \\
U \equiv |E_a|^{2} - |E_b|^{2}, \\
V \equiv |E_r|^{2} - |E_l|^{2}.
\end{matrix} </math>

from the Jones vectors to the corresponding Stokes vectors; more convenient forms are given below. The map takes its image in the cone defined by |''I'' |<sup>2</sup> = |''Q'' |<sup>2</sup> + |''U'' |<sup>2</sup> + |''V'' |<sup>2</sup>, where the purity of the state satisfies ''p'' = 1 (see below).

The next figure shows how the signs of the Stokes parameters are determined by the helicity and the orientation of the semi-major axis of the polarization ellipse.

[[File:StokesParamSign1.png|center]]

===Representations in fixed bases===
In a fixed (<math>\hat{x},\hat{y}</math>) basis, the Stokes parameters when using an ''increasing phase convention'' are

:<math> \begin{align}
I&=|E_x|^2+|E_y|^2, \\
Q&=|E_x|^2-|E_y|^2, \\
U&=2\mathrm{Re}(E_xE_y^*), \\
V&=-2\mathrm{Im}(E_xE_y^*), \\
\end{align}
</math>

while for <math>(\hat{a},\hat{b})</math>, they are

:<math> \begin{align}
I&=|E_a|^2+|E_b|^2, \\
Q&=-2\mathrm{Re}(E_a^{*}E_b), \\
U&=|E_a|^{2}-|E_b|^{2}, \\
V&=2\mathrm{Im}(E_a^{*}E_b). \\
\end{align}
</math>

and for <math>(\hat{l},\hat{r})</math>, they are

:<math> \begin{align}
I &=|E_l|^2+|E_r|^2, \\
Q &=2\mathrm{Re}(E_l^*E_r), \\
U & = -2\mathrm{Im}(E_l^*E_r), \\
V & =|E_r|^2-|E_l|^2. \\
\end{align} </math>