Difference between revisions of "Stokes parameters"

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\section{(Optional)}
 
\section{(Optional)}
In case you are rusty on your complex number manipulation, we calculate explicitly $P_U$ and $P_V$.
+
In case you are rusty on your complex number manipulation, we explicitly calculate $P_U$ and $P_V$.
 +
 
 +
Calculating $P_U$:
  
 
\begin{equation}
 
\begin{equation}
Line 71: Line 73:
 
\end{equation}
 
\end{equation}
  
Calculating $P_U$ by substitution:
 
 
\begin{align}
 
\begin{align}
 
     P_U &= \langle E_a^2 \rangle - \langle E_b^2 \rangle \\
 
     P_U &= \langle E_a^2 \rangle - \langle E_b^2 \rangle \\
Line 78: Line 79:
 
     &= \langle E_x E_y^* \rangle + \langle E_x ^* E_y \rangle \\
 
     &= \langle E_x E_y^* \rangle + \langle E_x ^* E_y \rangle \\
 
\end{align}
 
\end{align}
where we use the linearity of expectation in the last line. And lo and behold- it matches the definition from earlier!!!
+
where we use the linearity of expectation in the last line.
 +
 
 +
Calculating $P_V$:
  
 
\begin{equation}
 
\begin{equation}
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\end{equation}
 
\end{equation}
  
Calculating $P_V$ by substitution:
 
 
\begin{align}
 
\begin{align}
 
     P_V &= \langle E_l^2 \rangle - \langle E_r^2 \rangle \\
 
     P_V &= \langle E_l^2 \rangle - \langle E_r^2 \rangle \\

Revision as of 20:16, 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 Examples

5 (Optional)

In case you are rusty on your complex number manipulation, we explicitly calculate and .

Calculating :

where we use the linearity of expectation in the last line.

Calculating :