Hold a (linear) polarizing film or lens up to a light, and the light’s intensity will decrease by half.
Most polarizing lenses are made of parallel chains of polymers, so when electromagnetic radiation hits this sheet of chains, it excites electrons in the polymers to vibrate along the chains, and the electric field that gets transmitted is the component that is parallel to the chains.
From an unpolarized light source where the incident radiation has electric field vectors oriented in random directions, there will be approximately equal amounts of vector components oriented parallel and perpendicular to the polarizer’s axis (the direction of the polymer chains). Thus, only half of the total amount of electric field per unit of time through the area of the sheet gets transmitted, and this quantity - electric field x time x area - corresponds to the intensity of light.
But once you’ve got light that’s polarized along one axis, what if you pass it through another linear polarizer? If this second polymer sheet is oriented in the same direction as the first, then everything that got through the first one will get through the second one. It’ll be as if there were only one polarizer where the incident light is concerned. Introduce an angle between the two axes of polarization, and now the linearly polarized light from the first polarizer will get split into components parallel and perpendicular to the axis of the second, and only the parallel components will get transmitted. Intensity is proportional to the square of the electric field amplitude, so if θ is the angle between the polarization axes, then
Itransmit = (E||/Etotal)2 Iincident = (Etotal cos(θ)/Etotal)2 Iincident = cos2(θ)I_incident
This can be observed experimentally if you take two linear polarizers and rotate one against the other - you will see the light go from about half intensity of the surroundings to completely dark and back again. (Watch at :50)
At theta = π/2, the polarizing axes are perpendicular to each other, and the light that is transmitted through the first polarizer has no chance of passing through the second one. But if you insert a third polarizer between these two, it is possible to transmit light through all three polarizers. This is because as long as the radiation that is incident on the last polarizer is not polarized perpendicularly to the axis of polarization, light will be transmitted because there will be a component of the phase vector that is parallel to the polarizing axis. Since the middle polarizer can be oriented to change the polarization of the incoming light (any orientation besides totally perpendicular or totally parallel will work), you will be able to pass light through three polarizers even when the first and last and oriented so that alone they would completely block the transmission of light.
Experiments such as these demonstrate the wave nature of light; the wave model for electromagnetic radiation is based on Maxwell’s equations (or more compactly the Riemann-Sommerfeld equation) which today forms the basis of the classical optics. It was a major development after geometrical (ray-based) optics, which is still a useful model for studying light propagation through lenses, and is juxtaposed with/a precursor to quantum optics and nonclassical light, which employs the particle nature of light (as photons).
Linear polarization of light also serves as an illuminating analogy for the Stern-Gerlach experiment of quantum mechanics…