====== Retardance of optical elements ======


===== Characterizing the retardance in fibers =====

Hui and O'Sullivan \cite{Hui.OSullivan:2009} in section 4.5 discuss the measurement of polarization mode dispersion (PMD) of fibers. There are two types of //birefringence// in an optical fiber geometrical birefringence (inherited in the geometry of the drawn fiber) and stress birefringence due to non-symmetric stress. Then there is also external influences on the fiber, that lead to polarization dispersion.

Several methods for measuring the PMD, one of them being the Jones matrix method (4.5.6) and one the Mueller matrix method. (4.5.7)

===== Measuring the properties of a wave plate =====
The problem is similar to measuring a fiber, so chapter 4.5.6 and 4.5.7 are interesting.

==== Mueller Matrix method ====
This is based on \cite{Hui.OSullivan:2009}. As a reminder, the Mueller matrix, applied to a normalized stokes vector $[s_{1,in}, s_{2,in}, s_{3,in}]$ looks like 
$$ \begin{bmatrix} s_1 \\ s_2 \\ s_3 \end{bmatrix}  = \begin{bmatrix} m_{11} & m_{12} & m_{13} \\ m_{21} & m_{22} & m_{23} \\ m_{31} & m_{32} & m_{33} \end{bmatrix} \begin{bmatrix} s_{1,in} \\ s_{2,in} \\ s_{3,in} \end{bmatrix} $$

If $\mathbf{s}_i$ is the basis vectors with a 1 on position $i$, all three output stokes vectors, will form the Mueller matrix.

Also two independent input polarization states (no need to be orthogonal) allow determining the Mueller matrix as follows.
Measure $\mathbf{s}_a$ and $\mathbf{s}_b$ for two independent input polarizations. $\mathbf{s_1} = \mathbf{s_a}$ and then we can find a orthogonal vector to $\mathbf{s_1}$ and $\mathbf{s_b}$ through the cross-product. $$ \mathbf{s_3} = \mathbf{s_1} \times \mathbf{s_b} $$
The third vector can then be found by using the orthogonality through $$\mathbf{s_2} = \mathbf{s_3} \times \mathbf{s_1} $$.