# Difference between revisions of "Stokes parameters"

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STOKES PARAMETERS

## 1 Notation and Conventions

In the following, remember that $E_{x}$ and $E_{y}$ are COMPLEX numbers! We will use the “physicist notation" when talking about complex numbers. In particular, recall that for some complex number $z=x+iy$ , where $x,y\in {\mathcal {R}}$ , the complex conjugate is defined as $z^{*}=x-iy$ and the squared magnitude is given by $z^{2}=zz^{*}=x^{2}+y^{2}$ .

There’s also two different conventions on describing right vs. left hand polarization. Here we will use the definition where counterclockwise is right, and clockwise is left.

## 2 Definition

Stokes parameters are used to describe the polarization state of EM radiation. There are 4 Stokes parameters: $P_{I}$ is the total intensity, $P_{Q}$ is the polarization along the coordinate axes, $P_{U}$ is the polarization along the $45^{\circ }$ line between the coordinate axes, and $P_{V}$ is circular polarization.

{\begin{aligned}P_{I}&=\langle E_{x}^{2}\rangle +\langle E_{y}^{2}\rangle \\P_{Q}&=\langle E_{x}^{2}\rangle -\langle E_{y}^{2}\rangle \\P_{U}&=\langle E_{x}E_{y}^{*}\rangle +\langle E_{x}^{*}E_{y}\rangle \\P_{V}&=i(\langle E_{x}E_{y}^{*}\rangle -\langle E_{x}^{*}E_{y}\rangle )\end{aligned}}\,\! ## 3 Changing Bases

Let’s switch bases to understand these definitions a little better. Denote the Cartesian basis $({\hat {x}},{\hat {y}})$ , the $45^{\circ }$ rotated Cartesian basis $({\hat {a}},{\hat {b}})$ , and the circular basis $({\hat {r}},{\hat {l}})$ , which are defined as follows:

{\begin{aligned}({\hat {a}},{\hat {b}})&={\Big (}{\frac {{\hat {x}}+{\hat {y}}}{\sqrt {2}}},{\frac {{\hat {x}}-{\hat {y}}}{\sqrt {2}}}{\Big )}\\({\hat {r}},{\hat {l}})&={\Big (}{\frac {{\hat {x}}+i{\hat {y}}}{\sqrt {2}}},{\frac {{\hat {x}}-i{\hat {y}}}{\sqrt {2}}}{\Big )}\end{aligned}}\,\! and

{\begin{aligned}E&=E_{x}{\hat {x}}+E_{y}{\hat {y}}\\&=E_{a}{\hat {a}}+E_{b}{\hat {b}}\\&=E_{l}{\hat {l}}+E_{r}{\hat {r}}\end{aligned}}\,\! Note that ${\hat {x}}$ , ${\hat {y}}$ , ${\hat {a}}$ , ${\hat {b}}$ , ${\hat {l}}$ , and ${\hat {r}}$ are all unit vectors. From these definitions, it also follows that

{\begin{aligned}E_{a}&={\frac {E_{x}+E_{y}}{\sqrt {2}}}\\E_{b}&={\frac {E_{x}-E_{y}}{\sqrt {2}}}\\E_{l}&={\frac {E_{x}-iE_{y}}{\sqrt {2}}}\\E_{r}&={\frac {E_{x}+iE_{y}}{\sqrt {2}}}\end{aligned}}\,\! By plugging and chugging these definitions, we see that

{\begin{aligned}P_{I}&=\langle E_{x}^{2}\rangle +\langle E_{y}^{2}\rangle \\&=\langle E_{a}^{2}\rangle +\langle E_{b}^{2}\rangle \\&=\langle E_{l}^{2}\rangle +\langle E_{r}^{2}\rangle \\P_{Q}&=\langle E_{x}^{2}\rangle -\langle E_{y}^{2}\rangle \\P_{U}&=\langle E_{a}^{2}\rangle -\langle E_{b}^{2}\rangle \\P_{V}&=\langle E_{l}^{2}\rangle -\langle E_{r}^{2}\rangle \end{aligned}}\,\! Notice that all the Stokes parameters are real numbers. It might not be obvious because of all the complex conjugation, but it is easy to prove- thus it is left as a simple exercise for the reader :) A hint on how do this: define $E_{x}=a+ib$ and $E_{y}=c+id$ , where $a$ , $b$ , $c$ , $d$ $\in {\mathcal {R}}$ . Using these definitions, calculate the Stokes parameters- you’ll see that you only end up with real values.

## 4 (Optional Complex Conjugation Example)

In case you are rusty on your complex number manipulation, we explicitly calculate $P_{U}$ and $P_{V}$ .

Calculating $P_{U}$ :

{\begin{aligned}P_{U}&=\langle E_{a}^{2}\rangle -\langle E_{b}^{2}\rangle \\&={\Big \langle }{\Big (}{\frac {E_{x}+E_{y}}{\sqrt {2}}}{\Big )}{\Big (}{\frac {E_{x}^{*}+E_{y}^{*}}{\sqrt {2}}}{\Big )}{\Big \rangle }-{\Big \langle }{\Big (}{\frac {E_{x}-E_{y}}{\sqrt {2}}}{\Big )}{\Big (}{\frac {E_{x}^{*}-E_{y}^{*}}{\sqrt {2}}}{\Big )}{\Big \rangle }\\&={\frac {1}{2}}\langle E_{x}^{2}+E_{x}E_{y}^{*}+E_{x}^{*}E_{y}+E_{y}^{2}\rangle -{\frac {1}{2}}\langle E_{x}^{2}-E_{x}E_{y}^{*}-E_{x}^{*}E_{y}+E_{y}^{2}\rangle \\&=\langle E_{x}E_{y}^{*}\rangle +\langle E_{x}^{*}E_{y}\rangle \\\end{aligned}}\,\! where we use the linearity of expectation in the last line.

Calculating $P_{V}$ :

{\begin{aligned}P_{V}&=\langle E_{l}^{2}\rangle -\langle E_{r}^{2}\rangle \\&={\Big \langle }{\Big (}{\frac {E_{x}-iE_{y}}{\sqrt {2}}}{\Big )}{\Big (}{\frac {E_{x}^{*}+iE_{y}^{*}}{\sqrt {2}}}{\Big )}{\Big \rangle }-{\Big \langle }{\Big (}{\frac {E_{x}+iE_{y}}{\sqrt {2}}}{\Big )}{\Big (}{\frac {E_{x}^{*}-iE_{y}^{*}}{\sqrt {2}}}{\Big )}{\Big \rangle }\\&={\frac {1}{2}}\langle E_{x}^{2}-iE_{x}^{*}E_{y}+iE_{x}E_{y}^{*}+E_{y}^{2}\rangle -{\frac {1}{2}}\langle E_{x}^{2}+iE_{x}^{*}E_{y}-iE_{x}E_{y}^{*}+E_{y}^{2}\rangle \\&=i(\langle E_{x}E_{y}^{*}\rangle -\langle E_{x}^{*}E_{y}\rangle )\end{aligned}}\,\! 