Polarisation in Detail

13.1 Jones Calculus

The Jones vector represents the polarisation state of a monochromatic plane wave:

$\mathbf{E} = \begin{pmatrix} E_x \\ E_y \end{pmatrix} = \begin{pmatrix} A_x\,e^{i\delta_x} \\ A_y\,e^{i\delta_y} \end{pmatrix}$

Optical elements are represented by $2 \times 2$ matrices:

• Linear polariser at angle $\theta$: $\mathbf{P}(\theta) = \begin{pmatrix}\cos^2\theta & \sin\theta\cos\theta \\ \sin\theta\cos\theta & \sin^2\theta\end{pmatrix}$

• Quarter-wave plate (fast axis horizontal, retardation $\pi/2$): $\mathbf{Q} = \begin{pmatrix}1 & 0 \\ 0 & e^{i\pi/2}\end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & i\end{pmatrix}$

• Half-wave plate (retardation $\pi$): $\mathbf{H} = \begin{pmatrix}1 & 0 \\ 0 & e^{i\pi}\end{pmatrix} = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$

Theorem 13.1. The output of a sequence of optical elements is the product of their Jones matrices applied to the input Jones vector:

$\mathbf{E}_{\mathrm{out} = \mathbf{M}_n \cdots \mathbf{M}_2\,\mathbf{M}_1\,\mathbf{E}_{\mathrm{in}}}$

13.2 Stokes Parameters

For partially polarised light, the Stokes parameters are:

$S_0 = |E_x|^2 + |E_y|^2, \quad S_1 = |E_x|^2 - |E_y|^2, \quad S_2 = 2\,\mathrm{Re}(E_x E_y^*), \quad S_3 = 2\,\mathrm{Im}(E_x E_y^*)$

The degree of polarisation is

$P = \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0}$

For fully polarised light: $P = 1$. For unpolarised light: $S_1 = S_2 = S_3 = 0$.

13.3 Worked Example: Polarisation by Multiple Reflections

Problem. Unpolarised light is incident on a stack of $N$ glass plates at the Brewster angle. Find the degree of polarisation of the transmitted light.