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.
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:
Note that , , , , , and are all unit vectors. From these definitions, it also follows that
By plugging and chugging these definitions, we see that
4 A Visual Example
Sometimes the Stokes parameters are combined into a vector, cleverly called the Stokes vector. The Stokes vector is simply .
From top to bottom:
Linearly polarized, horizontal:
Linearly polarized, vertical:
Linearly polarized, +45 degrees:
Linearly polarized, -45 degrees:
Circularly polarized, right handed:
Circularly polarized, left handed:
All the polarizations have been drawn on different axes to better illustrate how they can be decomposed.
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.
5 (Optional Complex Conjugation Example)
In case you are rusty on your complex number manipulation, we explicitly calculate and .
where we use the linearity of expectation in the last line.