So what is circularly polarized light?

Well, in the last video we talked about linearly polarized light.

In other words, the direction of the electric field doesn't change as you go throughout space.

So let's say that this is the propagation direction of our plane wave and then we've got our coordinate axis.

A linearly polarized light, if it's polarized initially in X, so it's got some X component, that X component will change throughout space.

So it'll get bigger and smaller and bigger and smaller because this is a sinusoidal wave, this is a plane wave.

So we expect the magnitude to go up and down as we go throughout space.

But it doesn't change direction.

The direction stays the same.

It's always pointed along the X axis.

Now, circularly polarized light doesn't behave this way.

It actually changes the direction that this arrow is pointing as you go throughout space.

So this is our Z axis.

Now, you might ask, how on earth is that possible?

That seems just bizarre.

And the answer has to do with the wave nature of light and the fact that we can... add two polarized waves together.

So let's say we have this polarized wave, which is polarized in the X direction.

Let's say we also want to add a wave that's polarized in the Y direction.

So this is our Y axis.

But let's say that instead of having the maximum of the Y polarization match up with the maximum of the X polarization, so that would look something like this, instead of doing that, why don't we offset it?

So let's say that the maximum of our Y polarization happens when our X polarization is equal to zero.

So this is the maximum of our Y polarization.

And similarly, it decays to zero, or the sine wave goes to zero when the X polarization is at a maximum.

So if we were to trace that out in the Y axis, it would look something like this.

So we've got arrows pointing along the Y axis, and then our arrows start pointing backwards.

And most importantly, at this point, so let's call this Z equals zero, we have an entirely X polarized wave, or what looks like purely X polarized light.

So if we were to add up these two polarizations, this is X and this is Y, then initially, our wave is just polarized in X.

And I'm going to draw this total in a different color, let's do blue.

So initially, this is our total electric field, it's polarized in X.

But as we go along some distance, our total field ends up being polarized in Y.

And then if we go some distance more, it's polarized in negative X, and then negative Y, and then X, and so on and so on.

And if we were to trace out what this looks like, this would actually be, if they have the same magnitude, a circle.

So I've done my best to draw the, this is what it would look like, where we've got, initially we're in front of the axis, then we go behind the axis, and we sort of curve around, we continually curve around this axis.

And this is what's known as right-hand circularly polarized light.

So if you stuck your thumb out in this direction, and you were to curve your fingers of your right hand, you'd get that they travel in the same direction as this wave curves around the axis.

And so here's a visualization more of what this looks like, this curving light, or this curving polarization.

Notice it doesn't change its magnitude, so it stays constant in magnitude, it doesn't go from zero to one, it just changes its direction.

So you can kind of see how it curves to the right, or sort of a right-handed helix.

And if we advance the time, we can watch this wave propagate as you increase the time.

So we can see it's propagating down the Z-axis, and the electric field is sort of curving out this circle as we propagate it.

And if we freeze the helix in time, we can see that it's a right-handed helix, so it follows the curve of our right-handed fingers as we point our thumb in the direction of propagation.

So let's get rid of that.

So that's fine and good, but how do we mathematically model this?

And the answer is going to be with our Jones vectors.

So how do we represent a wave that's shifted, that's delayed with respect to the other wave by a quarter of the wavelength?

So this distance would be lambda over four, this distance here, whoop, what the fuck?

So this distance is lambda over four, our wavelength over four.

This total distance, a full half period, would be lambda over two, and this would be our full wavelength, so we've completed a full cycle.

So when we write down our X-polarized wave, we can write out a formula for that, we know what it is, and then we write down our Y-polarized wave, we need to delay this wave by lambda over four.

So we need to set Z to Z minus lambda over four.

And let's see what happens when we do that.

So our X-polarized wave, we can just write as, let's say, some magnitude E-naught in the X direction, e to the j omega t minus kz, and now our Y-polarized wave, wow, that looks like an X, that's actually a Y, our Y-polarized wave has the same magnitude, and we're assuming that, so this is what you need to assume for circularly polarized light, e to the j omega t minus k, and now we said we need to replace Z with Z minus lambda over four, so Z minus lambda over four, and we can rewrite that, so E-naught Y e to the j omega t minus kz, plus, so what's k times lambda over four?

Well, k is two pi over lambda times lambda over four, lambdas cancel, and we get pi over two, so plus pi over two.

And now the really clever part, you might notice we can just take this out of our wave, so we can factor this out front, and we'll have E-naught e to the j pi over two times our traveling wave that we had before, omega t minus kz, and this remembers the part that we're not interested in, so at least not after this point, at least not after this point, because we know that it's a traveling wave, we know that it's traveling in the Z direction, we only care about the components, so if we just write down the X and the Y components into our lazy man's vector, which is almost a Jones vector, it's just E-naught X plus E-naught e to the j pi over two Y, now we could write this in column vector form, and factor out the E-naught, and in that case we'll just have one, and e to the j pi over two, which is the imaginary number I, and so for the Jones vector, we just drop the amplitude, and we normalize the vector, so you need to divide by the square root of two, and this is our Jones vector for right-hand polarized light, and this contains all the information that we had before, but it's super, super simple, so rather than writing down this plus this, we just write down this super simple column vector, and this will actually allow us to do some really cool things, now I should make one note on conventions, so I've been using the convention that a traveling wave is represented as e to the j omega t minus kz, there's some people that do e to the j kz minus omega t, and in this case right-hand polarized light would be one minus I, there are also different conventions on whether this is right-hand polarized, or left-hand polarized, and the convention that I follow, and what seems to be the most common one, is the handedness convention, so stick your right hand out, curve your fingers, if your fingers follow the path of the light, then the light is right-hand polarized, if it follows the path of your left hand, it's left-hand polarized, so if instead we had shifted the wave, instead of shifting the y polarized wave in this direction, we could have also shifted it in this direction, and that would have given us left-hand polarized light, which the Jones vector, we could just represent as one minus I, and so these are two circularly polarized forms of light, and they correspond to helices, that are either right or left-handed, and are propagating along the z direction, but you might wonder, what if we hadn't shifted by a perfect lambda over four, what if we had shifted by a little bit less, or a little bit more, and that's a perfectly reasonable question, and we'll answer that in the next video, on elliptically polarized light, and you'll get the same exact thing, if you instead had different amplitudes, in front of this x and y, but we'll go over that in the future video, so I hope you enjoyed this one, if you did please give it a like down below, and subscribe to my channel, also if you have any questions or comments, please feel free to post those down below, and I'll try to get back to you as soon as I can, and thanks for watching, I'll see you next time.