The experimental set up used to characterize waveplates is shown schematically in the figure below.


In the above diagram lets define y to be out of the page and x to be in plane with the page. Here we start with purely vertical light, from the output of the first PBS. This light is then passed through the test waveplate, rotating the polarization as described by the Jones matrix we found in the theory section. The light is then projected back onto the x y basis by the second PBS. The Intensity of the light at each port is then measured by a photo-diode or power meter. Port A is measuring the y projection and port b is measuring the x projection.

The Jones vector describing the light after waveplate is given by.

\begin{align} J^{'} = M^{'} \left( \begin{array}{c} 0 \\ 1\end{array} \right) = \left(\begin{array}{c} cos\theta sin\theta -cos\theta sin\theta e^{-i\phi_R} \\ sin^2\theta + cos^2\theta e^{-i\phi_R} \end{array} \right) \end{align}

The Jones vector describes the electric field, however in practice we will measure the intensity of light at each port. Thus the intensity measured at port B will be proportional to the square of the first element in J', similarly the intensity measured at port A will be proportional to the square of the second element. As a function of waveplate angular position $\theta$ the Intensity observed at each port is given by.

\begin{align} I_A = A |sin^2\theta + cos^2\theta e^{-i\phi_R}|^2 \end{align}
\begin{align} I_B = A |cos\theta sin\theta -cos\theta sin\theta e^{-i\phi_R}|^2 \end{align}

Where A, is some arbitrary constant.

In practice, as the waveplate will be arbitrarily mounted in its rotation stage. The angle to the optical axis $\theta \rightarrow \theta +\theta_o$, where $\theta_o$ is given by the placement of the waveplate rotation stage. Given the above expressions for the intensity at each port, measured intensities, as a function of angular position, can be fit where A, $\theta_o$, and
$\phi_R$ are the only free parameters.

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