Polarisation in Detail

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:

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

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: $S_1 = S_2 = S_3 = 0$.

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.

A systematic reference for standard Jones matrices:

The Poincaré sphere provides a geometric representation of polarisation states. The Stokes parameters $(S_1, S_2, S_3)$ normalised by $S_0$ are the Cartesian coordinates of a point on a unit sphere:

• North pole ($S_3 = 1$): right circular polarisation
• South pole ($S_3 = -1$): left circular polarisation
• Equator ($S_3 = 0$): linear polarisation (orientation varies with longitude)
• Intermediate latitudes: elliptical polarisation

Problem. Unpolarised light passes through a linear polariser at $0^\circ$, then a quarter-wave plate with fast axis at $45^\circ$, then a linear polariser at $90^\circ$. Find the transmitted intensity.

Some materials exhibit optical activity: the plane of linear polarisation rotates as light propagates. This arises from circular birefringence — different refractive indices for left and right circularly polarised light.

The rotation angle is $\phi = \alpha L$ where $\alpha$ is the specific rotation and $L$ is the path length.

In the Faraday effect, a magnetic field along the propagation direction induces circular birefringence:

$\phi = V B L$

where $V$ is the Verdet constant. Faraday rotation is non-reciprocal: reversing the propagation direction doubles the rotation, unlike natural optical activity which cancels upon reflection.

1. 3D cinema (IMAX): Projectors use orthogonal polarisation states for left and right eye images; polarising glasses separate them.
2. LCD displays: Liquid crystals rotate the polarisation of light; crossed polarisers convert rotation into intensity modulation.
3. Stress analysis (photoelasticity): Transparent materials under stress become birefringent; viewing through crossed polarisers reveals stress patterns.
4. Radar and remote sensing: Polarimetric radar measures the full Stokes vector of backscattered radiation to identify terrain and targets.
5. Quantum cryptography: The BB84 protocol encodes qubits in the polarisation states of single photons (horizontal/vertical and diagonal/anti-diagonal bases).
6. Optical isolators: Combining a polariser with a Faraday rotator creates a non-reciprocal device that allows light to pass in one direction only.