Optical · Experiment

Polarization & Malus's Law

Light's electric field can point in any transverse direction. A polarizer lets through only one direction — so chaining filters at different angles dims, blocks, or surprisingly revives the beam.

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About this experiment ›

What you are looking at

A beam of light travels left to right through a row of polarizing filters. The little arrows between the filters show the direction the light's electric field is oscillating — its polarization — and the brightness of the beam shows how much light is getting through. A meter at the right reads the transmitted intensity.

Polarization and the first filter

Ordinary light from the Sun or a bulb is unpolarized: its electric field jitters in every transverse direction at random (shown as a starburst of arrows). A polarizer only transmits the component of the field along its transmission axis, so after the first filter the light is linearly polarized along that axis — and exactly half the intensity gets through, since on average only half the random field lies along the axis:
I₁ = ½ I₀  (unpolarized → first polarizer)

Malus's law

Once the light is polarized, a second filter (the "analyzer") set at angle Δθ to the first transmits only the projected component, and intensity goes as the cosine squared:
I = I_in · cos²(Δθ)
Line the axes up (Δθ = 0) and all the light passes; cross them at 90° and the cosine is zero — total darkness. Rotate the analyzer and watch the brightness follow the cos² curve plotted below.

The three-polarizer surprise

Here is the famous twist. Two crossed polarizers (0° and 90°) block everything. Now slip a third polarizer at 45° between them — and light reappears! Each stage only needs cos²(45°) = ½, so the chain transmits ½ · ½ · ½ = ⅛ of the original (12.5%). The middle filter "re-projects" the polarization partway around, letting some survive the final filter. Try the 3-polarizer preset to see it. This is impossible to explain if you think a polarizer simply "removes" light — it actively redefines the polarization direction.

Things to try

Sweep Polarizer 2 from 0° to 90° and watch Malus's law dim the beam to nothing. Set up crossed polarizers, then add the third in the middle and rotate it — transmission peaks when it sits at 45°. Polarizers like these are how LCD screens, polarized sunglasses, and photographers' filters control light.