This is the second in a series of articles about Quantum Cryptography. The first article is here. At the end of the 19th century, people thought that they had fully understood light. They had showed that it underwent diffraction and interference - classically wave-like phenomenon. Maxwell's equations predicted electromagnetic waves that travelled at, you guessed it, the speed of light. This applet showing diffraction (the beams represent the directions of the maxima).
All was well in the the world of Physics.
When examining 'Black Body' radiation, treating light as if it were a wave did not explain the spectrum of light emitted by a hot object. This was called the 'ultra-violet catastrophe' as the prediction was drastically incorrect at shorter wavelengths. The solution for this was to treat light as if it could only come in discrete little packets, which were called photons. This is much easier to see by thinking about the photoelectric effect. (Applet for the photoelectric effect)
How does this square with polarisation? If light comes in photons, each photon can either go through the filter or not.
For simplicity, let's assume that a vertically polarised photon will go through a vertically polarised filter 100% of the time. This isn't quite true as there will be some 'ordinary' photon loss due to passing through material, but let's separate the effects.
The probability of the same photon passing through a horizontal filter is zero.
What about an intermediate angle?
Well, classically, a 45° filter will let through half of the power in the beam of light (or √2 of the amplitude). In terms of photons, this means that half of the photons are getting through. How is half related to 45°? Well cos 45° is 1/√2 - so.... the probability of a vertically polarised photon getting through the filter is cos2 θ where θ is the angle of the polaroid to the vertical.
There's more than this. If a photon passes through a filter, the filter affects the polarisation, so that photon now has a 50/50 chance of getting through a vertical polaroid, and 50/50 chance of getting through a horizontal polaroid. With just a horizontal polaroid, no light would get through. This can mean that if we have two crossed polaroids blocking light, then inserting a third can let more light through. This was discussed in classical terms in the previous article in the series.
The key things here is that a photon has a probability of getting through a polaroid filter which depends upon the angle of the filter. Each filter 'resets' the polarisation of any photon which emerges from it, so it doesn't matter how that photon 'started' as far as subsequent filters are concerned.