Editing Stokes parameters (section)

==Properties==
For purely [[monochromatic]] [[Coherence (physics)|coherent]] radiation, it follows from the above equations that

:<math>
Q^2+U^2+V^2 = I^2,
</math>

whereas for the whole (non-coherent) beam radiation, the Stokes parameters are defined as averaged quantities, and the previous equation becomes an inequality:<ref>H. C. van de Hulst ''Light scattering by small particles'', Dover Publications, New York, 1981, {{ISBN|0-486-64228-3}}, page 42</ref>

:<math>
Q^2+U^2+V^2 \le I^2.
</math>

However, we can define a total polarization intensity <math>I_p</math>, so that

:<math>
Q^{2} + U^2 +V^2 = I_p^2,
</math>

where <math>I_p/I</math> is the total polarization fraction.

Let us define the complex intensity of linear polarization to be

:<math>
\begin{align}
L & \equiv  |L|e^{i2\theta} \\
 & \equiv  Q +iU. \\
\end{align}
</math>

Under a rotation <math>\theta \rightarrow \theta+\theta'</math> of the polarization ellipse, it can be shown that <math>I</math> and <math>V</math> are invariant, but

:<math>
\begin{align}
L & \rightarrow  e^{i2\theta'}L, \\
Q & \rightarrow  \mbox{Re}\left(e^{i2\theta'}L\right), \\
U & \rightarrow  \mbox{Im}\left(e^{i2\theta'}L\right).\\
\end{align}
</math>

With these properties, the Stokes parameters may be thought of as constituting three generalized intensities:

:<math>
\begin{align}
I & \ge  0, \\
V & \in  \mathbb{R}, \\
L & \in  \mathbb{C}, \\
\end{align}
</math>

where <math>I</math> is the total intensity, <math>|V|</math> is the intensity of circular polarization, and <math>|L|</math> is the intensity of linear polarization. The total intensity of polarization is <math>I_p=\sqrt{|L|^2+|V|^2}</math>, and the orientation and sense of rotation are given by

:<math>
\begin{align}
\theta &= \frac{1}{2}\arg(L), \\
h &= \sgn(V). \\
\end{align}
</math>

Since <math>Q=\mbox{Re}(L)</math> and <math>U=\mbox{Im}(L)</math>, we have

:<math>
\begin{align}
|L| &= \sqrt{Q^2+U^2}, \\
\theta &= \frac{1}{2}\tan^{-1}(U/Q). \\
\end{align}
</math>