Total reflection in Feynman picture

[Swamp Thing]

In his popularized version of QED (Strange Theory...) Feynman explains reflection from an air-water boundary (including total internal reflection) by summing up the amplitudes of various paths that bend at the boundary but are straight everywhere else. As we add more contributions, we see a Cornu spiral that gives us a large amplitude for the actual classical path. (Fig. 29 in my edition).

He does say this is a simplified picture, because it assumes straight lines that change direction only at the boundary.

My question is, can we relax that assumption a little bit, and still get to the correct result? We don't want to go totally non-classical by considering arbitrarily curved paths, but we can include more various straight line paths with a single break / bend anywhere in the media.

So let's consider a bundle of bent straight lines where some bend just below the boundary, and some bend above the boundary. So we're allowing the bending to happen anywhere within the media, and not necessarily at the boundary. If we sum these paths, will they add up correctly and work as if the bending was effectively at the boundary?

 

[kuruman]

When I toyed with diagrams and constructions as Feynman suggested in this wonderful little book, I applied them to reflection as follows. First I convinced myself through the "Cornu spiral" diagrams that the straight line between two points A and B is the path of maximum probability. Having that, I treated reflection as finding the path of maximum probability when light goes from A to B with the constraint that it passes through point C situated on a plane surface (the reflector). The exercise is to find the point C that maximizes the probability. The paths that I used were straight lines from A to C and C to B because non straight line paths would only decrease the probability as I previously established. So I moved point C along the intersection of the reflector plane and the plane defined by lines AC and CB, then did the Cornu spiral bit and verified the law of specular reflection. Does this answer your question?

 

[PeterDonis]

So we're allowing the bending to happen anywhere within the media, and not necessarily at the boundary. If we sum these paths, will they add up correctly and work as if the bending was effectively at the boundary?

No.

The exact rule is that you add up the amplitudes for all the paths--not just straight lines, not just line segments with bends in between, but arbitrary curved paths, paths that go all over the place but still end up at the desired end point. In principle you have to include paths that go out to the Moon and back, as long as they do it in the required time (note that the endpoints are not just places in space, but points in spacetime--places in space at particular instants of time). As I think Feynman notes, that means you have to allow for paths where light does not move at the speed of light.

Whenever you adopt an approximation that is not the above exact rule--i.e., whenever you leave out paths--you have to make sure you only leave out paths that make, when summed up, a smaller contribution to the total amplitude than the paths you include. In your suggested approximation, you are including paths that continue on a straight line across the boundary, but how do you know those paths make a larger contribution to the amplitude than paths that bend at the boundary without being reflected? Remember that we already know light passing across a boundary that is not reflected should be refracted. So if you include any paths that cross the boundary, you need to rethink your whole approach because you can no longer assume that a straight line path is the path of maximum probability. You can only assume that within a single medium.

 

[Swamp Thing]

you are including paths that continue on a straight line across the boundary, but how do you know those paths make a larger contribution to the amplitude than paths that bend at the boundary without being reflected?

At least in one aspect, moving the "knee" of the dogleg path vertically across the boundary seems to work as it should. (That may not mean it is valid in general, though). This seems to suggest that in some cases at least, the rule that bending occurs classically at the boundary may emerge naturally if we don't start with that assumption, but sum up a lot of dogleg paths with no prior assumption/knowledge of the boundary.

When we vary the bend point slowly from below to above the boundary, we get a nice Cornu spiral. And the inflection point (where it moves fastest) occurs exactly when the dogleg coincides with the Snell's Law path.

On the right, the dotted line is from Snell's law. The thick colored path is an arbitrary path. The spiral on the left was generated by sweeping the bend point along the Y axis.

In the video below (sorry for the poor quality) you can see the path changing slowly on the right, and the cornu spiral building up on the left. The red dot on the right shows the phase angle of the path. It changes its direction of rotation just when the path coincides with the dotted Snell path, and of course that is also the inflection point of the spiral.

 

[PeterDonis]

At least in one aspect, moving the "knee" of the dogleg path vertically across the boundary seems to work as it should.

You don't know this. You are still considering paths with only one bend. That is legitimate if the bend occurs at the boundary, because it is already known that within a single medium, the straight line path is representative of the total amplitude you would get from summing all of the paths of whatever shape. But if you put the bend somewhere else, you no longer know that. So you might be getting what seems like a "right" answer only because you are inadvertently leaving out other paths which would make contributions to the amplitude of the same magnitude as the one-bend paths you are including--but whose contributions, if included, would spoil the nice spiral you have found.

In other words, since you already know the right answer, it's tempting to think that any subset of paths you pick out that gives you that answer must be "right". But that's not necessarily the case. You have to show that you are including all the paths up to your chosen level of accuracy. You can't just assume that you are because you are getting an answer that looks right.

 

[mfb]

I would start much earlier. Take a single homogeneous medium, allow a single bend anywhere in 2 or 3 dimensions and see if you can get the right propagation out of that.
As far as I remember, even this simple looking problem fails in two dimensions, and works in three dimensions only.

 

[Swamp Thing]

I see your point. I suppose it's a case of making a number of big errors that cancel out by sheer luck in a particular situation. Interesting but highly misleading.

Or looking at it another way, if we make small perturbations around the known-good solution, that cluster of paths might well show a behavior that seems to guide us back towards the right direction, but this would be a terrible way to find an unknown solution from scratch.