Hard data on a hot topic: Studying Climate Change With Lasers

Understanding polarisation

Electromagnetic waves

Electromagnetic (EM) waves are transverse waves — the oscillations in the electric and magnetic field are directed perpendicular to the direction of wave propagation (unlike, for example, sound waves, which are longitudinal). If we consider an EM wave propagating along the z-axis in free space, then the electric and magnetic fields will be directed entirely in the x and y directions and will vary sinusoidally in space and time. The variation in space (at one particular time) is shown in the picture below:

The electric and magnetic fields in an EM wave in free space are orthogonal and coupled: if we know the form of one field we can immediately calculate the other. Since the electric field oscillations of an EM wave are many orders of magnitude larger in amplitude than the magnetic ones we typically describe a wave in terms of its electric field, as this field dominates the interactions of the EM wave with matter. The most general solution for the electric field of an EM wave of frequency $ω$ in free space, propagating along the z-axis with wavevector $k$ , can be written as

$\vec{E} = \left(\begin{array}{c} E_x \\ E_y\end{array}\right) = \left(\begin{array}{c} A_x \cos (\omega t-\delta_x-kz) \\ A_y \sin(\omega t-\delta_y-kz)\end{array}\right)\,,$

where $A x$ and $A y$ are amplitudes obeying the relation

$A_x^2 + A_y^2 = \vert\vec{E}\vert^2$ ,

and $δ x$ and $δ y$ are dimensionless phase shifts. The combination of amplitudes $A x$ , $A y$ and phases $δ x$ , $δ y$ associated with a given EM wave determine its polarisation.

Polarisation applet

How the amplitudes and phases in the equation for the EM wave's electric field determine the polarisation is best understood by looking at graphs of how the electric field varies in time at a given point — for example at the point z=0.

In the applet below, the x- and y-components of the electric field at z=0 are both plotted over one oscillation period. Below them, the information from these two graphs are combined in a Lissajous figure, which plots, as a line, all the places that the electric field vector points to during one oscillation period. Adjust the sliders below the graphs and look at the Lissajous figure, and you should quickly see why we talk of polarisation using terms like "linear", "elliptical", and "circular"...

E_x against time over one period

E_y against time over one period

Lissajous Figure: Plot of (E_x,E_y)
over an entire period

Use the following sliders to adjust the figures above:

A_x = 1.00

A_y = 0.00

δ_x = 0.00π

δ_y = 0.00π