Maxwell to Snell

Today was a really interesting day in my optics class. We’re doing chapter 9 in our text and I wanted to make sure that I motivated the material well. What’s weird about this text is that it waits until chapter 9 to do ray optics. This chapter has the lens equation, curved surfaces, etc. You know, all the stuff you can do in general physics. So why does it wait until chapter 9? Well, because we’re trying to ground everything in Maxwell’s equations for electric and magnetic fields. Go back and look at your general physics book and you’ll likely find that there’s a big gap between things like Ampere’s law and Snell’s law. I learned so much in class today that I wanted to make sure I wrote it down, so here you go.

We did Snell’s law back in chapter 3. I love that derivation, because we show that the boundary conditions that Maxwell’s equations enforce lead directly to three things:

where i stands for incident, r for reflected, and t for transmitted. It’s a really cool argument that comes down to this: If:

then

as long as can take on any value. If is time, then you get the equation above, which is the same as saying that the color going in is equal to the color going out. If the interface lies in a plane (which is crucial to this argument), then can represent one of the spatial dimensions in the plane and you get the law of reflection and Snell’s law. Very cool! A direct path from Maxwell to Snell.

But there’s a problem. If you don’t consider plane waves, you can’t do it! The math is critically dependent on both the idea that the wave is in the form of a plane and that the interface is a plane. Break down either of those, and you lose some of the valuable math tools we’ve put together.

So today was all about looking at non-plane waves. First we considered the notion of a plane wave interacting with a non-plane interface. Going into the interface is fine, but the reflected and transmitted waves would definitely not be plane waves, so we can’t assume angle in equals angle out and Snell’s law. One thing we talked about was the notion that any curved interface can just be zoomed in upon until the portion of the interface you’re considering looks flat. The thinking goes that then your picture looks like the plane wave one and everything’s good. But the big problem is that , acting as one of the spatial variables that span that mini-plane can’t take on any value, since the plane does not extend forever. Therefore we can’t use that way of thinking to say that reflection and refraction work there.

So what to do? Well, let’s go back to the beginning. For plane waves, we assumed that the field was of the form:

and we checked to see what conditions would be in place if we enforced Maxwell’s boundary conditions at the (planar) interface. Now we need to try a more general version of the electric field. The plane (given by the  part of the equation) represents the locus of points that all have the same phase. If we want to consider a more interesting, curvy surface, we should put an expression for that into the exponent and see what happens:

Where R(r) is a surface with contours representing lines of constant phase.

What we did was plug that into the wave equation (which comes, of course, from Maxwell’s equations) and made some assumptions of the small-ness of the wavelength compared to any changes of the material. After a lot of work (see section 9.1 of the text) we get to the Eikonal equation:

This finally tells us that to find the next contour, you should take a step perpendicular to your current contour, with your step size being proportional to 1/n (ie, the steeper the hill, the closer the contours are). So now at least we can start seeing how the wave evolves.

But that still doesn’t get us to pencil-thin rays (like lasers) propagating and hitting things like lenses etc. To get there, we need to do a little more math. We can derive Fermat’s idea of least time by playing around with the Eikonal equation. We find that the path a ray would take between two points is the one that would take the least amount of time. Then, from that idea, we can get to Snell’s law by asking how light would navigate going from one point in one medium to another in a different medium (this is a pretty standard Fermat-based homework problem).

Aha! Finally! We get Snell’s law (and, very similarly, the law of reflection) for zoomed in flat things that don’t have to extend to infinity. With that in hand, we can go off and do lenses and curved mirrors and have some fun (especially with ABCD matrices!).

So, here’s the path: Maxwell -> wave equation -> general form for a non-planar wave front -> assume small wavelengths -> Eikonal equation -> Fermat -> Snell. Awesome (at least I thought so).

What do you think? Here’s some starters for you:

1. I’m in this class and I thought this was cool. Connecting everything back to Maxwell has really connected some ideas for me.
2. I’m in this class and I thought this was a waste of time. I knew all this stuff worked because we’ve been using lasers, not plane waves, in lab all semester long.
3. I’m in this class but wasn’t able to make it today. Can I print this out and turn it in for a standard?
4. Could you, I don’t know, do some screencasts to fill in some of the gaps in the text?
5. I like this. I’ve always been frustrated with the typical approach in general physics because . . .
6. I think this is dumb. You’re making statements about the general derivation that aren’t true. We can get all these results quite easily and generally by . . .
7. Why do you do the Eikonal stuff? You’re going to be hitting abrupt interfaces like lenses that break the assumptions built in!
8. I like this because everything should be tied to Maxwell. How would you do that for . . .