Three polarizers, 45° apart

If light is passed through two polarizing filters before arriving at a target, and both of the filters are oriented at 90° to each other, then no light will be received at the target. If a third filter is added between the first two, oriented at a 45° angle (as shown below), light will reach the target.

Why is this the case? As I understand it, a polarized filter does nothing except filter out light--it does not alter the light passing through in any way. If two filters exist that will eliminate all of the light, why does the presence of a third, which should serve only to filter out additional light, actually act to allow light through?

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"As I understand it, a polarized filter does nothing except filter out light--it does not alter the light passing through in any way." This is not the case. – dmckee Apr 22 '13 at 19:08
@dmckee: Actually, I'd agree that it only filters out light -- but light at 45 degree polarization really is a (coherent) linear combination of vertically and horizontally polarized light. Projections need P_a P_b = 0 need not mean that P_a P_c P_b = 0. I guess one can quibble about "does not alter the light passing through in any way." – wnoise Apr 23 '13 at 1:02
It is critically important that after passing a polarizer the light has a new well defined polarization regardless of what it polarization state was before. That is a property of the light that has been changed. You can express it in a number of ways but you need to understand the change in order to explain how inserting a third polarize can result in light passing through where none had passed before. – dmckee Apr 23 '13 at 1:12

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.

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Exactly what I was looking for. Thank you! – Vaindil Apr 22 '13 at 19:05
It might be worth pointing out that, although one can explain the effect classically, one can also do the experiment with single photons. In that case, a quantum mechanical interpretation is needed. – hanno Jan 25 '14 at 0:37
@hanno very true! But, and forgive me if this sounds at all dismissive, I wish to leave my answer as is because it is more likely that a google searcher will be looking for a brief and succinct explanation to the classical effect. However, I do encourage someone to post a quantum mechanical explanation; that would definitely round out this question nicely. – Jim Jan 26 '14 at 14:24
@Jimnosperm - a year and some later, I added that second explanation. Of sorts. – Floris Apr 10 '15 at 22:27
I have read the link in this answer and must say I am not really happy with it. It just postulates that the filter can "take the projection of the polarization vector" without explaining how that happens. What does the filter actually do to the wavefront that could have this effect? I understand the QM explanation, but the classical explanation seems to me to be no explanation at all, at least as it is presented here. – Thriveth Apr 3 at 4:21

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.

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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.

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• quantum experiments are probabilistic (weird non-classical thing):
• they take a mixture of states
• the result of the experiment is one of the states
• each result occurs with a probability proportional to the amplitude of the states
• after the experiment, the system assumes the state that was observed
• well known example: probability that an electron hits a given part of a wall in the double slit experiment
• when a photon hits a polarizer, that is an experiment, and two outcomes are possible:
• it passes through, and its new orientation is the same as that of the polarizer
• it does not pass through
• a physical property of polarizers that we take for granted: the probability that a photon passes is given by the angle between the polarization and the filter:
• parallel: 100%
• perpendicular: 0%
• but what about 45 degree polarized photon? We can conclude that from the horizontal and parallel ones:
• every photon polarized vertically is equivalent to a superposition of a 45 and -45 degree states with amplitude $\frac{1}{\sqrt{2}}$ each
• the 45 state always passes (parallel)
• the -45 degree state never passes (perpendicular)
• so the probability that the vertical photon will pass is $\frac{1}{\sqrt{2}}$, i.e. proportional to the 45 degrees one
• when light passes the vertical filter, all photons that pass are now in the vertical polarized state

• if those photons hit the horizontal filter immediately, 0% of them pass (perpendicular)
• if they hit a 45 degree filter first, $\frac{1}{\sqrt{2}}$ pass, and assume the 45 degree state.

Then they hit the horizontal filter, that one is also at 45 degrees from the previous state, so again $\frac{1}{\sqrt{2}}$ pass.

So the net number that passes is $\frac{1}{\sqrt{2}} \times \frac{1}{\sqrt{2}} = \frac{1}{2}$

• this is all very similar to multiple chained Stern Gerlach experiments

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