Difference between revisions of "Stokes parameters"

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\maketitle
 
\maketitle
  
\section{Stokes Parameters}
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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 $z^2 = z z^* = x^2 + y^2$.
\subsection{Notation and Conventions}
 
In the following, remember that $E_x$ and $E_y$ are COMPLEX numbers! We will use physicist notational conventions when talking about complex numbers. In particular, recall that for some complex number $z = x + iy$, where x and y are real numbers, 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.
  
\subsection{Definition}
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\section{Definition}
Light (or more generally, electromagnetic radiation,) can be described as a wave. Waves oscillate, and the orientation of these oscillations is called the \emph{polarization} of the wave. Light is often described as linearly polarized or circularly polarized.
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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.
 
 
Stokes parameters are used to describe the polarization state of electromagnetic 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 \\
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\end{align}
 
\end{align}
  
Notice that all the Stokes parameters are \emph{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, are real numbers. Using these definitions, calculate the Stokes parameters- you'll see that you only end up with real values.
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\section{Changing Bases}
 
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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:
\subsection{Changing Bases}
 
Let's switch bases to understand these definitions a little better; in particular, why $P_U$ is called the polarization along the $45^\circ$ line between the coordinate axes, and $P_V$ is called circular polarization. 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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\end{align}
 
\end{align}
  
By plugging and chugging these definitions, it's not too difficult to verify that the following is true:
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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 \\  
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\end{align}
 
\end{align}
  
So although it's not quite obvious from the complex definition, the names for the different P's make sense. Going into a different basis makes it easier to see.
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[[File:stokes1.png]]
 
 
\subsection{A Visual Example}
 
 
 
Sometimes the Stokes parameters are combined into a vector, cleverly called the Stokes vector. The Stokes vector $\vec{S}$ is simply  $\vec{S} = (P_I, P_Q, P_U, P_V)$.
 
 
 
The picture below illustrates, from top to bottom:
 
  
Linear polarization, horizontal: $\vec{S} = (1,1,0,0)$
+
Notice that all the Stokes parameters are \emph{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.
 
 
Linear polarization, vertical: $\vec{S} = (1,-1,0,0)$
 
 
 
Linear polarization, +45 degrees: $\vec{S} = (1,0,1,0)$
 
 
 
Linear polarization, -45 degrees: $\vec{S} = (1,0,-1,0)$
 
 
 
Circular polarization, right handed: $\vec{S} = (1,0,0,-1)$
 
 
 
Circular polarization, left handed: $\vec{S} = (1,0,0,1)$
 
 
 
All the polarizations have been drawn on different axes to better illustrate how they can be decomposed.
 
 
 
[[File:stokes1.png]]
 
  
\subsection{Optional: Complex Conjugation Example}
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\section{(Optional Complex Conjugation Example)}
 
In case you are rusty on your complex number manipulation, we explicitly calculate $P_U$ and $P_V$.
 
In case you are rusty on your complex number manipulation, we explicitly calculate $P_U$ and $P_V$.
  

Revision as of 08:04, 17 December 2017

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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

Stokes1.png

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 and , where , , , . 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 and .

Calculating :

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

Calculating :