Wave plate in the Poincaré sphere

Poincaré sphere representation of a Jones vector under the action of either a half- or a quarter-wave plate

Installation

Clone or download this reposository and run it using python wihtin a conda environment. If this your first time using conda, please consider installing anaconda.

Requirements

This code requires ipympl to dynamically interact with the Poincaré sphere. You can install ipympl in your conda environment via pip

pip install ipympl

or conda

conda install -c conda-forge ipympl

Convention

There are multiple conventions for the Jones vectors and matrices. However, we follow the Wikipedia Jones calculus convention because it is easily available to everyone.

The orthogonal polarisation vectors are:

$$ H = \begin{bmatrix}1 \cr 0\end{bmatrix} $$

$$ V = \begin{bmatrix}0 \cr 1\end{bmatrix} $$

$$ D = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \cr 1\end{bmatrix} $$

$$ A = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \cr -1\end{bmatrix} $$

$$ R = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \cr -i\end{bmatrix} $$

$$ L = \frac{1}{\sqrt{2}}\begin{bmatrix}1 \cr i\end{bmatrix} $$

For the half- and quarter-wave plates (HWP and QWP, respectively) we use the Jones matrix for an arbitrary linear phase retarder (LPR)

$$ LPR(\eta,\vartheta,\varphi) = \begin{bmatrix} \cos(\eta/2)-i\sin(\eta/2)\cos(2\vartheta) & -\sin(\eta/2)\sin(\varphi)\sin(2\vartheta)-i\sin(\eta/2)\cos(\varphi)\sin(2\vartheta) \cr \sin(\eta/2)\sin(\varphi)\sin(2\vartheta)-i\sin(\eta/2)\cos(\varphi)\sin(2\vartheta) & \cos(\eta/2)+i\sin(\eta/2)\cos(2\vartheta)\end{bmatrix} $$

with $\eta = \pi$ ($\eta = \pi/2$) for a HWP (QWP). The angle $\varphi$ identically vanishes for both a HWP and a QWP, but we keep it to introduce the action of a liquid crystal in a future release.

Effect of a wave plate on an arbitrary Jones vector

We assume an initial Jones vector of the form

$$ \begin{bmatrix}\cos(\theta/2) \cr e^{i\phi}\sin(\theta/2)\end{bmatrix} $$

The action of $LPR(\eta,\vartheta,\varphi)$ on this initial Jones vector leads to

$$ LPR(\eta,\vartheta,\varphi)\begin{bmatrix}\cos(\theta/2) \cr e^{i\phi}\sin(\theta/2)\end{bmatrix}=\begin{bmatrix}\alpha \cr \beta\end{bmatrix} $$

with

$$ \alpha = [\cos(\eta/2)-i\sin(\eta/2)\cos(2\vartheta)]\cos(\theta/2)+[-\sin(\eta/2)\sin(\varphi)\sin(2\vartheta)-i\sin(\eta/2)\cos(\varphi)\sin(2\vartheta)]e^{i\phi}\sin(\theta/2) $$

$$ \beta = [\sin(\eta/2)\sin(\varphi)\sin(2\vartheta)-i\sin(\eta/2)\cos(\varphi)\sin(2\vartheta)]\cos(\theta/2)+[\cos(\eta/2)+i\sin(\eta/2)\cos(2\vartheta)]e^{i\phi}\sin(\theta/2) $$

To represent Jones vectors in the Poincaré sphere we calculate the Stokes parameters as

$$ S_1 = |\alpha|^2-|\beta|^2 $$

$$ S_2 = 2\mathrm{Re}(\alpha\beta^*) $$

$$ S_3 = 2\mathrm{Im}(\alpha\beta^*) $$

where $^*$ is the complex conjugate.

Screenshots

A HWP at 22.5° transforms H into D:

A QWP at 0° transforms R into D:

Suggested exercises

1. Verify that a HWP at an angle $\vartheta$ rotates an arbitrary vector in the Poincaré sphere by 180° around an axis on the equator at $2\vartheta$ w.r.t. the "H" axis.

2. Verify that a QWP at an angle $\vartheta$ rotates an arbitrary vector in the Poincaré sphere by 90° around an axis on the equator at $2\vartheta$ w.r.t. the "H" axis following the left-hand rule (palm pointing towards the initial vector, thumb parallel to the QWP rotation axis).

3. Verify that the wave plate angles to produce the orthogonal polarisation vectors – up to a global phase – in a PBS $\to$ QWP $\to$ HWP configuration are:

Vector	QWP	HWP
H	0°	0°
V	0°	45°
D	0°	22.5°
A	0°	-22.5°
R	-45°	0°
L	45°	0°

4. Verify that the wave plate angles to project the orthogonal polarisation vectors onto the transmission port of a PBS in a QWP $\to$ HWP $\to$ PBS configuration are:

Vector	QWP	HWP
H	0°	0°
V	0°	45°
D	45°	22.5°
A	45°	-22.5°
R	0°	22.5°
L	0°	-22.5°