So what do we get from that?

We have now some idea that the time dependence of

the dielectric response relates to this refractive index change,

this dispersion that we've seen before.

That's one good thing. But what we really need

for going forward now in the shape of light is the plane wave,

because we are going to learn a lot

about light and how it propagates just from a plane wave.

And the plane wave is the solution to the wave equation.

So we need to derive from Maxwell's equations the wave equation.

This is a derivation that

any electrical engineering student that works with

light should probably be able to do on the back of a napkin.

It's a really simple thing,

and it connects what you're doing to Maxwell's equations.

So the steps are,

we take Faraday's law,

which related E to the time derivative of B,

and we take the curl of both sides.

And we do that because the curl B,

of course, is Ampere's Law.

We substitute M for B mu naught H,

and it's only mu naught because

the relative permeability of materials and optical frequencies is almost always one.

And then, we use Ampere's law to substitute from del cross H into D. And, finally,

the other constitutive relationship from D and epsilon naught

epsilon E. I've written it down here before an anisotropic case.

But if you don't like that,

just get rid of the whole vector and tensor signs.

So, the point is we use both curl equations and both constitutive relations.

So, we have pretty much used all of Maxwell's equations

because the divergence versions of Maxwell's equations are if you remember,

included in the curl equations.

So, now we've eliminated the magnetic field to

the point we head to first order equations relating

E to B and D to H. And now we have a single second order,

second derivatives right here.

Equation relating E to E. Now we're going to go monochromatic.

So, you already have monochromatic.

So we just rearrange this equation and just like we use

the resonant frequency of the dipole earlier on,

we're going to recognize that some of these terms here are of

the vacuum wave number two pie over the free space wavelength.

This is the vector wave equation with it's anisotropic turn right there.

Monochromatic, there are no time derivatives,

so is the frequency domain.

And most of the time we'll just worry about scalar waves here.

So we'll get rid of this tensor,

we'll call these vector electric fields just scalars.

We use a vector identity on curl curl and we get to the scalar simplification.

And if you take del squared here,

this is just in all three dimensions,

the second spatial derivatives.

So the simplest version of this is the second derivative of the electric field.

Let's say there's step to Z is K squared E. So that's your classic simple wave equation.

That's the equation we're solving in this whole course.

And the whole point of the last few lectures is simply

you should know how that came from Maxwell's equations,

and what approximations or not working to it.