Three polarizers, 45° apart

This link: http://alienryderflex.com/polarizer/ has an excellent explanation; much better than anything I could write here.

Essentially, it says that this occurs because the 45 degree filter outputs a projection of the vertical rays at 45 degrees. This, in turn, has a horizontal component, which the final filter projects in its output.

This answer was written for another question that was deleted a few minutes ago. I decided to post it here even though the effect it describes duplicates Floris' answer:

Photons passing through a medium don't just punch their way through like bullets. They are absorbed by the atoms of the medium and then re-emitted. (Incidentally, the reduction in speed for light passing through a medium is caused by each photon "orbiting" the nucleus of an atom before being re-emitted. They travel at c, but the distance travelled is greater.)

Light passing through a vertically oriented polarization filter emerges with a classical mechanical wave polarization in the vertical direction. If the light then passes through a horizontal filter, 100% of the classical mechanical wave action is eliminated.

But if you insert a filter oriented at 45 degrees between the vertical and horizontal filters, you introduce an element of quantum probability into the apparatus. It creates a quantum effect, and you can witness the quantum probabilistic transmission of light.

All the photons passing through the vertical filter are vertically oriented UNTIL they pass through the filter oriented 45 degrees. When they're absorbed and re-emitted by the 45 degree filter, 50% are vertically oriented, and 50% are horizontally oriented, as the quantum effect allows photons to be EITHER up or down, vertical or horizontal. 45 degrees is not allowed, but as 45 degrees is 50% of the angle between the vertical and horizontal filters, the emissions from that filter are half vertical and half horizontal.

The horizontal filter then emits only the horizontally oriented classical mechanical wave action that has passed through the 45-degree filter.

A year later, here is a probabilistic (pseudo QM) explanation.

I am confused by the diagram that appears to show unpolarized laser light - I thought that most lasers by their nature produce polarized light; after the first polarizer that question is moot, so let's start there.

A polarized photon can be thought of as being in a mixture of states - when it approaches a polarizer, it's either parallel or perpendicular (it either passes, or it doesn't). When you polarize a photon and then immediately "test" it with another polarizer at right angles, you know the state it's in: it is "perpendicular" to the second polarizer and will be stopped.

But when you have another angle, you have a certain probability of passing or not passing. In particular, when you're coming in polarized at 45°, there is an equal chance of passing or not passing (it is in a mixture of two states, if you like). So half the photons will pass the second polarizer - and they will come out "rotated".

These photons now hit the third polarizer - again, at 45 degrees. Again, you have a 50-50 chance that such a photon is parallel, and is passed on.

We therefore have a 1 in 4 chance of passing the two polarizers, where before we had none.