Wave polarisation

The polarisation of a wave describes its direction of oscillation. The polarisation vector $\hat{n}$ describes the plane of vibration. There are two primary cases:

• A longitudinal wave has the displacement from equilibrium along the direction of propagation, i.e., $\hat{n}=\hat{k}$.
• A transverse wave has the displacement perpendicular to the direction of propagation, i.e., $\hat{n}\cdot \hat{k}=0$. The image below shows transverse polarised waves.¹

Basics

We describe polarisation in terms of a few core polarisation states, that describe the trajectory of the field vector over time at a fixed plane perpendicular to the direction of travel. For a wavefunction:

$$\vec{E}(\vec{r},t)=\left(E_{x,o}{e}^{i{\phi }_x}\hat{x}+E_{y,o}{e}^{i{\phi }_y}\hat{y}\right)\exp \left[i(kz-\omega t+\phi )\right]$$

• Linear polarisation is when the time evolution of $\vec{E}$ is a line on an $xy$-plane, assuming a $\hat{z}$ propagation direction.
  • This happens when ${\phi }_y-{\phi }_x=m\pi$, for an integer $m$.
• Circular polarisation is when the time-evolution trace is a circle on the $xy$-plane.
  • This happens when $E_{x,o}=E_{y,o}$ and ${\phi }_y-{\phi }_x=\frac{\pi }{2}+m\pi$.
• Elliptical polarisation happens in the remainder of cases.
  • i.e., this happens in cases not covered above.

Both circular and elliptical polarisation are split into left/right-handed polarisation. We determine it primarily with the phase difference $\Delta \phi ={\phi }_y-{\phi }_x$.

• If positive, it is left-hand polarised.
• If negative, it is right-hand polarised.

We usually also describe polarisation with Jones vector, which are 2D vectors that succinctly describe wave polarisation.

Devices

An optical device that polarises light is called a polariser. A basic type of polariser is the linear polariser, which filters out light waves that aren’t aligned with its transmission axis. The resulting light is linearly polarised in a single direction.

A few important parameters:

• The principal transmittance is the fraction of incident light parallel to the polariser’s transmission axis that is passed through.
• The minor transmittance is the fraction of incident linear light perpendicular to the transmission axis that is passed (or leaked) through by the polariser.
• The extinction ratio is the ratio of minor transmittance to principal transmittance.

See also

• Dielectric polarisation, which describes the alignment of molecular dipoles within a material in response to an applied electric field

Footnotes

1. From Introduction to Electrodynamics, by David J. Griffiths. ↩