Difference between revisions of "Stokes parameters"

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\maketitle
 
\maketitle
  
FIXME: seems to be different from wikipedia. check on that. (i.e. right vs left polarization, a vs b).
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\section{Notation and Conventions}
 
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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 $z^2 = z z^* = x^2 + y^2$.
\section{A Note on Notation}
 
In the following, remember that $E_x$ and $E_y$ are COMPLEX numbers!!! We will use the ``physicist notation" when talking about complex numbers. 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 $z^2 = z z^* = x^2 + y^2$.
 
  
 
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.
 
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.
  
 
\section{Definition}
 
\section{Definition}
Stokes parameters: used to describe polarization state of EM radiation.  
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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{align}
 
\begin{align}
 
     P_I &= \langle E_x^2 \rangle + \langle E_y^2 \rangle \\
 
     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_Q &= \langle E_x^2 \rangle - \langle E_y^2 \rangle \\
     P_U &= \langle E_x E_y^* \rangle + \langle E_y E_x^* \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_y E_x^* \rangle )
+
     P_V &= i (\langle E_x E_y^* \rangle - \langle E_x^* E_y \rangle )
 
\end{align}
 
\end{align}
These 4 $P$s are the 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.
 
  
 
\section{Changing Bases}
 
\section{Changing Bases}
We could just leave it at that, and take these definitions as given. Every definition has a story, so let's understand it. However, a physical picture is not really obvious to me from these definitions. This is not very obvious (in my opinion) just from inspection of these formulas. Let's break this down further, and switch bases to make this a little more immediate. 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:
+
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{align}
 
\begin{align}
 
     (\hat{a},\hat{b}) &= \Big(\frac{\hat{x} + \hat{y}}{\sqrt{2}}, \frac{\hat{x} - \hat{y}}{\sqrt{2}} \Big) \\
 
     (\hat{a},\hat{b}) &= \Big(\frac{\hat{x} + \hat{y}}{\sqrt{2}}, \frac{\hat{x} - \hat{y}}{\sqrt{2}} \Big) \\
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     &= E_l \hat{l} + E_r \hat{r}
 
     &= E_l \hat{l} + E_r \hat{r}
 
\end{align}
 
\end{align}
Note that $\hat{x}$, $\hat{y}$, $\hat{a}$, $\hat{b}$, $\hat{l}$, and $\hat{r}$ are all unit vectors. From these definitions, we can solve for $E_a$, $E_b$, $E_l$, and $E_r$ in terms of $E_x$ and $E_y$:
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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{align}
 
\begin{align}
 
     E_a &= \frac{E_x + E_y}{\sqrt{2}} \\
 
     E_a &= \frac{E_x + E_y}{\sqrt{2}} \\
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\end{align}
 
\end{align}
  
It can now be shown that
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By plugging and chugging these definitions, we see that
 
\begin{align}
 
\begin{align}
 
     P_I &= \langle E_x^2 \rangle + \langle E_y^2 \rangle \\  
 
     P_I &= \langle E_x^2 \rangle + \langle E_y^2 \rangle \\  

Revision as of 20:11, 12 December 2017

Course Home

STOKES PARAMETERS

1 Notation and Conventions

In the following, remember that and are COMPLEX numbers! We will use the “physicist notation" when talking about complex numbers. In particular, recall that for some complex number , where , the complex conjugate is defined as and the squared magnitude is given by .

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: is the total intensity, is the polarization along the coordinate axes, is the polarization along the line between the coordinate axes, and is circular polarization.

3 Changing Bases

Let’s switch bases to understand these definitions a little better. Denote the Cartesian basis , the rotated Cartesian basis , and the circular basis , which are defined as follows:

and

Note that , , , , , and are all unit vectors. From these definitions, it also follows that

By plugging and chugging these definitions, we see that

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 and , where , , , . Using these definitions, calculate the Stokes parameters- you’ll see that you only end up with real values.

4 Proof

Here we explicitly do out the math (in case you don’t believe the above and are too lazy to do it yourself.)

Calculating by substitution:

where we use the linearity of expectation in the last line. And lo and behold- it matches the definition from earlier!!!

Calculating by substitution:

5 Examples