# Difference between revisions of "Stokes parameters"

STOKES PARAMETERS

## 1 Notation and Conventions

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

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: ${\displaystyle P_{I}}$ is the total intensity, ${\displaystyle P_{Q}}$ is the polarization along the coordinate axes, ${\displaystyle P_{U}}$ is the polarization along the ${\displaystyle 45^{\circ }}$ line between the coordinate axes, and ${\displaystyle P_{V}}$ is circular polarization.

{\displaystyle {\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 ${\displaystyle ({\hat {x}},{\hat {y}})}$, the ${\displaystyle 45^{\circ }}$ rotated Cartesian basis ${\displaystyle ({\hat {a}},{\hat {b}})}$, and the circular basis ${\displaystyle ({\hat {r}},{\hat {l}})}$, which are defined as follows:

{\displaystyle {\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

{\displaystyle {\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 ${\displaystyle {\hat {x}}}$, ${\displaystyle {\hat {y}}}$, ${\displaystyle {\hat {a}}}$, ${\displaystyle {\hat {b}}}$, ${\displaystyle {\hat {l}}}$, and ${\displaystyle {\hat {r}}}$ are all unit vectors. From these definitions, it also follows that

{\displaystyle {\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

{\displaystyle {\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}}\,\!}

Things to notice:

• All the Stokes parameters are real numbers. It might not be obvious because of all the complex conjugation, but it is easy to prove it to yourself. It is left as a simple exercise for the reader. For example, to do this, define ${\displaystyle E_{x}=a+ib}$ and ${\displaystyle E_{y}=c+id}$, where ${\displaystyle a}$, ${\displaystyle b}$, ${\displaystyle c}$, ${\displaystyle d}$ ${\displaystyle \in {\mathcal {R}}}$. Using these definitions, calculate the Stokes parameters- you’ll see that you only end up with real values.

## 5 (Optional)

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

Calculating ${\displaystyle P_{U}}$:

${\displaystyle E_{a}={\frac {E_{x}+E_{y}}{\sqrt {2}}},\,E_{b}={\frac {E_{x}-E_{y}}{\sqrt {2}}}\,\!}$
{\displaystyle {\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 ${\displaystyle P_{V}}$:

${\displaystyle E_{l}={\frac {E_{x}-iE_{y}}{\sqrt {2}}},\,E_{r}={\frac {E_{x}+iE_{y}}{\sqrt {2}}}\,\!}$
{\displaystyle {\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}}\,\!}