Tilt your head while wearing polarised sunglasses and look at the glare off a wet road or a car bonnet — it brightens and dims as you rotate. Look at a phone screen through them and turn the phone: at one angle it blacks out entirely. Something about the light has a direction baked into it, a direction that has nothing to do with where the light is going. That hidden property is polarisation, and it is a direct fingerprint of the fact that light is a transverse wave: its electric field oscillates across the line of travel, and which way it points across that line is a real, physical, measurable thing.

This page has one idea, in two halves. First, the states of polarisation — linear, circular, elliptical — and what physically distinguishes them. Second, the astonishingly tidy bookkeeping that handles them: Jones calculus, in which every polarisation state is a two-component vector and every optical element (a polariser, a wave plate) is a 2×2 matrix. Sending light through a stack of elements becomes nothing more than multiplying matrices. Polarisation optics turns into linear algebra — and that is a gift.

What polarisation is

A light wave travelling along z has an electric field lying in the xy-plane, perpendicular to the direction of travel. At each instant that field is a vector — it has an x-part and a y-part — and polarisation is the story of how that vector's tip moves as the wave goes by.

• Linear polarisation: the two components rise and fall exactly in step, so the tip traces a straight line. The field always points along one fixed direction (horizontal, vertical, or any angle between).
• Circular polarisation: the components are equal in size but a quarter-cycle (90^\circ) out of step. While one is at its peak the other is at zero, so the tip sweeps out a circle, rotating once per wave period — left-handed or right-handed depending on the sign of the phase lag.
• Elliptical polarisation: the general case — unequal components, or any other phase difference — traces an ellipse. Linear and circular are just its two special limits.

Ordinary light from the Sun or a bulb is unpolarised: a jumble of countless little wave trains with every orientation, changing far too fast to pin down. A polariser is a filter that passes only the field component along its transmission axis and absorbs the rest — which is exactly how the glare-killing sunglasses work.

The states, drawn

Imagine looking straight down the beam, back toward the source, watching only the tip of the electric-field vector in the xy-plane. Each polarisation state draws a different shape. Step through the four cases:

Notice how naturally each state is captured by a pair of numbers: how much x, how much y, and the phase between them. That pair of (complex) numbers is exactly the Jones vector — the whole motive for the calculus we build next.

Jones vectors: a state in two complex numbers

Write the polarisation as a column of two complex numbers — the amplitude and phase of the x- and y-components:

The relative phase between the two entries is what encodes the shape; an overall phase and an overall size don't change the polarisation, so we usually normalise so that |E_x|^2 + |E_y|^2 = 1 (that sum is the intensity). The standard states:

Jones matrices: optics as multiplication

Here is the payoff. Every polarisation element transforms the input Jones vector into an output one by a 2×2 matrix M: \mathbf{J}_{\text{out}} = M\,\mathbf{J}_{\text{in}}. And to send light through element M_1 then M_2 then M_3, you just multiply the matrices — M_3 M_2 M_1 — in the order the light meets them, right to left. The whole apparatus collapses into a single matrix.

Feed 45^\circ linear light \tfrac{1}{\sqrt2}\binom{1}{1} into a quarter-wave plate \binom{1\ 0}{0\ i} and out comes \tfrac{1}{\sqrt2}\binom{1}{i} — circularly polarised light. That is how every pair of "3-D glasses" and every circular polariser is built: a linear polariser glued to a quarter-wave plate.

Malus's law

Send linearly polarised light of intensity I_0 through a polariser (an "analyser") whose axis makes an angle \theta with the light's polarisation. Only the component along the axis, E_0\cos\theta, gets through; intensity is amplitude squared, so

This is Malus's law, and it falls straight out of the Jones matrices. It says parallel polarisers (\theta = 0) pass everything, crossed polarisers (\theta = 90^\circ) pass nothing — total blackout — and at 45^\circ exactly half. Rotate the analyser and the transmitted intensity traces a smooth \cos^2 curve:

Here is the delightful twist that makes people gasp. Two crossed polarisers pass zero light. Now slip a third polariser at 45^\circ between them — and light comes through! Each stage passes \cos^2 45^\circ = \tfrac12 of what reaches it, so the output is I_0\cdot\tfrac12\cdot(\tfrac12)(\tfrac12) = \tfrac{I_0}{8} of the polarised input. Inserting a filter makes a dark system brighter — impossible for classical absorption, obvious once you know each polariser re-projects the state onto a new axis. It is a favourite demonstration and a genuine window onto how measurement works in quantum mechanics.

Worked examples

Example 1 — Malus's law. Vertically polarised light of intensity I_0 = 10\ \text{W/m}^2 hits an analyser at \theta = 30^\circ. The transmitted intensity is I = 10\cos^2 30^\circ = 10\,(0.866)^2 = 10(0.75) = 7.5\ \text{W/m}^2.

Example 2 — unpolarised then analysed. Unpolarised light of intensity I_0 first meets a polariser. Averaging \cos^2\theta over all angles gives \tfrac12, so I_0/2 emerges, now polarised. A second polariser at 60^\circ to the first then passes \tfrac{I_0}{2}\cos^2 60^\circ = \tfrac{I_0}{2}\,(0.5)^2 = \tfrac{I_0}{8}.

Example 3 — Jones multiplication. Horizontal light \binom{1}{0} through a horizontal polariser \binom{1\ 0}{0\ 0} gives \binom{1\ 0}{0\ 0}\binom{1}{0}=\binom{1}{0} — passes untouched. Through a vertical polariser \binom{0\ 0}{0\ 1} it gives \binom{0}{0} — total blackout, crossed polarisers, exactly as Malus's law says at 90^\circ.