# Stokes parameters

STOKES PARAMETERS

## 1 Stokes Parameters

### 1.1 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.

### 1.2 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}}\,\!}

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 ${\displaystyle E_{x}=a+ib}$ 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.

### 1.3 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}}\,\!}

### 1.4 A Visual Example

Sometimes the Stokes parameters are combined into a vector, cleverly called the Stokes vector. The Stokes vector ${\displaystyle {\vec {S}}}$ is simply ${\displaystyle {\vec {S}}=(P_{I},P_{Q},P_{U},P_{V})}$.

From top to bottom:

Linearly polarized, horizontal: ${\displaystyle {\vec {S}}=(1,1,0,0)}$

Linearly polarized, vertical: ${\displaystyle {\vec {S}}=(1,-1,0,0)}$

Linearly polarized, +45 degrees: ${\displaystyle {\vec {S}}=(1,0,1,0)}$

Linearly polarized, -45 degrees: ${\displaystyle {\vec {S}}=(1,0,-1,0)}$

Circularly polarized, right handed: ${\displaystyle {\vec {S}}=(1,0,0,-1)}$

Circularly polarized, left handed: ${\displaystyle {\vec {S}}=(1,0,0,1)}$

All the polarizations have been drawn on different axes to better illustrate how they can be decomposed.

### 1.5 (Optional Complex Conjugation Example)

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 {\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 {\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}}\,\!}