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Earlier in this course, we discussed linear polarization of electromagnetic radiation, and I demonstrate this at 75 megaHertz and at 10 gigaHertz.

Today, I will concentrate exclusively on the polarization of light, which is at a much higher frequency.

The light from the sun or light from light bulbs is not polarized.

So I can ask myself the question, now, what does it mean when light is not polarized?

Let's think of individual light photons as plane waves, with a well-defined direction of polarization.

So each one is linearly polarized.

A beam is coming straight out of the blackboard.

The first photon arrives, it's linearly polarized in this direction.

This second photon arrives, linearly polarized in this direction, so the electric field vector is oscillating like that.

Another photon, another photon, and another photon.

And what you see here, very clearly, that there is no preferred direction which you average over time, and that's what we call -- call unpolarized light.

It was Edwin Land who, in 1938, developed a material that can turn this into linearly polarized light, for which he became very famous, in addition to this demonstration that I showed you last time.

If I take one of Edwin Land's sheets, which will turn light into polarization in this direction, and I first take one photon, for instance, this one.

That one comes in from the blackboard towards you, and so here it is.

Oscillating the E vector like this, E0 is the maximum value of the electric field strength in that plane electromagnetic wave.

And this is the direction of the polarizer that I have through which this photon goes.

I can now make a simple calculation, by projecting this E-vector onto the preferred direction of polarization, and this new E-vector is now down by the cosine of theta, if this angle is theta, this E-vector is now E -- E0 times the cosine of theta.

If you ask me now, whether the light is reduced in intensity, I would have to say, "Yes, of course," because light intensity depends on the Poynting vector, and the pointing vector is always proportional to E0 squared, because the Poynting vector is the cross-product between E and B.

And if E is reduced, B is also reduced.

And so we get a cosine square reduction.

If, now, I average over all incoming photons -- so I take all of these, which represent an unpolarized beam -- so I get not only one like so, but I get one like so, and one like so, and one like so, and one like so -- then clearly, I have to calculate, now, the mean value of cosine square theta.

And the mean value of cosine square theta is one-half, and so if the intensity of the unpolarized beam, unpolarized light was originally I0, once it comes through this polarizer that Edwin Land gave me, then I get one-half I0, but that is now 100% polarized.

And it is 100% polarized in this direction.

And the one-half is the result of the average value of cosine square theta.

If this were the case, it would be an extremely ideal polarizer, we would call this an HN50 polarizer -- they don't exist, it's only in your head -- and the 50 refers to the fact that 50% get through polarized.

In the optics kits that we hand out today that we will need throughout this course, you don't have HN50 polarizers, they don't exist.

I don't quite know what yours is, I didn't measure it, yours may be an HN25 or maybe an HN30, which would then mean that the I0 strength of an unpolarized light of beam would not be half of I0, but maybe only .25, or .3.

But in any case, the light that will come through your linear polarizers will be very closely to 100% polarized.

So what I will do now, I will take unpolarized light, and I will have this light coming straight out of the blackboard perpendicular to you, with strength I0, and here is one of my polarizers, and the light that comes through here is linearly polarized in this direction.

And so we already know that one-half I0 will come through if it is an ideal polarizer, and it is polarized in this direction.

I take a second sheet, an identical one, I put it also in the plane of the blackboard, but I rotate it over an angle theta.

So here is now a second sheet, which has a preferred direction of polarization in -- in this direction, and the angle is rotated over an angle theta.

So between this one and this one is an angle theta.

And so you can now immediately tell what the intensity of the light is that comes through this second polarizer.

It must, of course, be polarized in this direction, because that is the allowed direction polarization for that second sheet -- and the intensity must now be one-half I0, because that's what comes in, and then I have to multiply it by the cosine square of theta.

I don't have to average it now over all angles, because there is only one value of theta between this sheet and this sheet, so this is now the new intensity, and it's all polarized in this direction.

And this law, whereby the light intensity is reduced by the factor cosine square theta, is known as Malus' Law.

Malus' Law.

If theta were 30 degrees, the light intensity here would be one-half I0 times the cosine square of 30 degrees, which is 0.75.

If theta were 0 degrees, that means that this sheet is in the same direction as this one, if everything were ideal, one-half I0 would get through the second sheet.

If theta is 90 degrees, then nothing will get through, because the cosine of 90 degrees is 0.

We call that crossed polarizers.

If you cross them like this, no light will get through.

Now, before I demonstrate this, I have to be honest with you, because the idea of reducing the energy of individual photons by reducing their electric field strength, as I did, is a cheat.

A light photon has a well-defined energy which depends uniquely on the frequency of the light.

Blue light has a higher frequency than red light, so blue light has a higher energy than red light.

And when you send blue light through a polarizer, the way I did here, it either comes through or it doesn't come through.

But if it does come through, it is still blue light, there is no such thing as a reduction in energy.

Whereas this reduction, by cosine theta, would imply that the energy goes down, and that moo- would imply, then, that there would be a color change, that it would no longer be blue.

And that's not the case.

If you want to treat this properly, you have to do it in a quantum mechanical way.

The interesting thing is that if you use quantum mechanics, you find exactly the same law, you find also Malus' Law.

So the law is OK, even though the derivation is not kosher.

Now, I want you to get out of your envelope one of your green plates, which is a linear polarizer.

This is the kind of plate that you have, you have three in there.

Only take one out.

These two lights shining on me, unpolarized light.

So the light that comes to you now is unpolarized.

I'm now going to hold in front of my face this polarizer.

So the light that comes through is linearly polarized in this direction.

And you are going to play the role of the second polarizer.

Close one eye, put the polarizer in front of your eye, and rotate it.

And you will see a huge difference in light intensity.

If you cross-polarize with me, then you can't see me.

That may make you very happy.

But keep in mind, if you can't see me, then I can't see you, either.

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