And don't be afraid, it doesn't involve any integrals, it's really just very simple vectors and 1 by 2 vectors and 2 by 2 matrices.

So let's say that I have a plane wave, so I'm just going to it's moving in, let's call this the z-direction, that would make this the x-direction and this the y-direction.

If we know the plane wave is moving in the z-direction, there's a bunch of ways the electric field could be pointing.

It could be pointing upwards in the x-direction, and in that case the h-field would be pointing this way in the in the y-direction, well let me label that x as y.

It could also be pointing that way in the y-direction, and then the h-field would be pointing in the minus x-direction, or it could be pointing somewhere in between.

So it could be pointing in the xy, anywhere really in the xy-plane, so anywhere in this plane, and then the h-field is just 90 degrees away from that.

Now one of the many beautiful things about plane waves is that the electric field and the magnetic field are just related by a constant eta, and this was the wave impedance of free space.

This only depends on the material that we're propagating through, so if we know what the e-field is, we automatically know what the h-field is, and similarly we know its direction because it has to be orthogonal to the electric field, and e cross h is pointing in the direction of propagation, so it's our pointing vector.

So knowing one of the fields of a plane wave fully determines the other, and so we're only going to worry about the electric field.

We're just going to study the electric field.

This is all we need to worry about in order to fully understand polarization.

So how do we go about representing that polarization?

So let's say we know the direction of propagation of the plane wave, and let's say it's along the z-axis.

How many degrees of freedom do we have to deal with?

Well, the electric field, like we just said, can be anywhere in this plane, so let's pretend now that it's pointing just straight up.

It's pointing in the in the x-direction.

How would we represent that in just mathematically?

And let's say it's got the amplitude e-naught.

Well, the electric field is a vector, and it's pointing in the x-hat direction.

It's got amplitude e-naught, and then it's a traveling wave in the z-direction, so we're gonna write that in complex notation e to the j omega t minus kz.

And how would we represent it if it were pointing in the y-direction?

So maybe it has amplitude e-naught in the y-direction.

Well, then the electric field is just equal to y-hat times e-naught e to the j omega t minus kz.

And in general, if it's pointing in some other direction, so it's got some component along x, so let's call that e-naught x, and some component along y, e-naught y, then we can write the total electric field as the x component, e-naught x, e to the j omega t minus kz, plus the y component, e-naught y, e to the j omega t minus kz.

Now notice that this e to the j omega t minus kz, this traveling wave component of the of the plane wave, is getting sort of redundant.

So the only real information that we're representing with our polarization is the x-hat part and the y-hat part.

So how much is pointing in x, how much is pointing in y?

We don't really care that the wave is traveling because we already knew that.

So we might just be lazy and say that the electric field is just equal to x-hat e-naught x plus y-hat e-naught y.

And if you were a lazy person that decided to do this, you'd be doing exactly what Jones did when he invented his calculus a long time ago, I think in the late 1900s.

And so this is a vector.

We can also write this as a column vector with just e-naught x and e-naught y.

And this is kind of cute because we have a three-dimensional problem, but because we assumed the direction of propagation, we only really have two degrees of freedom.

We only, our electric field can only be x and y.

So if we know that it's a traveling wave, we know its frequency, we can fully capture all of the information just in this vector, which is really cool.

You might also be wondering, well why is the electric field amplitude even important here?

Because all we care about is how much is contributed, or how much of the electric field is in the x direction, and how much of the electric field is in the y direction.

So this is sort of the super compactness of the electric field.

Say it's pointing only in the x direction, so its equation, its mathematical description would be x-hat e-naught e to the j omega t minus kz.

The Jones vector for that, or the vector that we would we would use if we weren't being super lazy, was just e-naught zero.

So it's just got some component in the x direction.

But if we were to divide this by the total amplitude, so let's divide this by 1 over e-naught, we would get the Jones vector, which is just 1, 0.

And this is a normalized vector, so it's got length 1, and it just tells us in what direction the electric field is pointing.

So if we had an electric field pointing in the y direction, or a traveling wave in the y direction, our Jones vector would be 0, 1.

If it were pointing half in the x direction, half in the y direction, 45 degrees, then this would be 1 over root 2, 1, 1.

And that's just, this is just a normalized vector pointing at 45 degrees in x and y.

Now all of these vectors are what's known as linearly polarized, and that just means that the electric field is pointing in some direction.

And if you were to advance the plane wave, it would still be pointing in the same direction.

So take our x example from before.

If we take one snapshot of the plane wave, which is pointing in, going in this direction, our electric field is pointing in the x-hat direction, so x, y, z.

And if we advance the plane wave, so we take it some time further, the electric field is in general going to have a different amplitude, because maybe this was at the maximum of the plane wave, so maybe this is slightly lower.

So let's say, where's, let's draw our E0 vector pointing downward now.

So this is E0, but it's still pointing in the x direction.

This is y, this is x.

It's still pointing in the x direction.

And so as the field, as we go in the z direction, the electric field stays pointing in the same direction.

So if we just draw what it looks like over all of the space between one plane that I've outlined and the other, this is what the electric field will trace out on its path in the z direction.

This is what it'll look like if we freeze it in time, so at t equals 0.

And if we let it go in time, then it'll move forward, so the wave, the phase front will advance and the wave will go, all the electric fields will shift in space.

And so this is linearly polarized light.

And it doesn't have to be pointing just in the x direction to be linearly polarized.

It could be pointing in the y direction, it could be pointing somewhere in between, it could be pointing anywhere in this two-dimensional plane.

The thing that makes it linearly polarized is that it stays the same direction as it advances.

And you might be asking, that seems really weird.

Why would it be able to change directions as it goes forward?

And that's going to be the subject of the next video, and circularly polarized light, which is where things get really interesting and where Jones vectors start to become really, really valuable.

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