polarization of light explained
The electric field of light shakes at right angles to the direction of travel. A polarizer is a sieve that lets that shaking through in only one direction. Overlap two of them at right angles and the light disappears, and slip one more between them and it comes back to life. Malus's law explains all of this with a single angle. The angle sets the brightness.
Light is a transverse wave, so the electric field shakes at right angles to the direction of travel. A polarizer lets through only the component along its transmission axis. The amplitude falls to A₀ cosθ, and the intensity is the square of the amplitude, so I = I₀ cos²θ. That is Malus's law. Polarizing sunglasses use the same principle. They filter, with a vertical transmission axis, the glaring light reflected horizontally off water or a road. Only the component that lies on the axis gets through.
Set down one more polarizer and turn the rear one, the analyzer. When the two axes are parallel, at 0 degrees, almost all the light passes, and when they are at right angles, at 90 degrees, it goes completely dark. The intensity that passes draws a cos²θ curve against the angle. A liquid-crystal display uses exactly this two-polarizer structure. The liquid crystal between them twists the direction of the light's shaking and sets the brightness of each pixel. The twist is the brightness of the pixel.
Two polarizers overlapped at right angles block the light completely. Slip a third plate at a slant between them, though, and the light comes back to life, because each plate projects the electric field onto a new axis. It is brightest at a middle angle of 45 degrees, and 1/8 of the original intensity gets through. If a polarizer only removed light, adding a plate should make things darker, yet it grows brighter, and that shows a polarizer is a device that realigns the direction of shaking onto a new axis. The plate you slip in brings the light back.
Natural light shakes evenly in every direction. After the first polarizer the intensity falls to exactly half no matter where the axis points, and from there on the light is polarized, lined up in one direction. From the next polarizer onward it follows Malus's law, cos²θ. Natural light becomes exactly half because every angle is mixed evenly, and averaging cos²θ over a full turn gives exactly 1/2. An even average makes the half.
The root of Malus's law is a plain projection. Drop the electric-field vector as a shadow onto the transmission axis and its length becomes the original times cosθ. The intensity is the square of that shadow, so it is cos²θ. Tilt the vector to 45 degrees and the shadow shrinks to about 0.707 of the original length, and the intensity, that square, is exactly half. Turn the vector and you can follow how the shadow shrinks and grows. The square of the shadow is the intensity.