[SOLVED] Refraction of light

[Fedda]

Ever since I was introduced to this concept in physics in school, it has bothered me, because I can't seem to understand it properly. Sure, I understand Snell's law and how the refractive index of different media affect lightwaves, but to paint the mental picture, all we were given were some horrible analogies, like two wheels connected on an axel, going from tarmac to sand/grass at an angle, changing their direction. Since then I have heard quite a few analogies, but none of them seem to satisfy me. I just cannot understand how a change in the phase velocity of light can make the individual photons change their directions. I have looked through several textbooks and web pages, but it doens't seem like any of them explain the phenomenon, they just simply state that 'light bends'. I guess (and hope) there's something simple that I'm just missing, and I'm looking like a fool here.

I don't know where else to look for answers than the forums for XKCD!

[jmorgan3]

Feynman gives a quantum-mechanical reason in section 26-6 here: http://www.physics.uci.edu/~demos/pdf/o ... t_time.pdf

[thoughtfully]

It wouldn't be the only property of light that's tricky to understand from a particle point of view. Just accept it as wavelike behavior, if those analogies work for you.

[Charlie!]

The individual photons are still waves. Quantum mechanics, eh? So if you understand why waves will change direction, you realio trulio understand why the photons change direction.

[Tass]

You've heard of the double slit experiment? You've heard the weird thing that each photon actually "goes through both slits". Well when light is refracting it is the same way, each photon in a way goes through every point in the lens, and then interferes with it self on the other side.

Or it might be better to think that light only becomes photons when we detect them, until then it is a wave with interferences and all that wave stuff.

Really to "fully" understand it you'd have to learn quantum mechanics.

[Fedda]

Yeah, we did the double slit experiment as well in physics recently, however just with the most basic maths associated with it. I see now how it's related to quantum mechanics. I guess I'll wait until college, so I can try to get a grasp on that as well (I know I probably won't, but just knowing that it's related to quantum mechanics makes the not knowing-part a little bit better)

[Fedda]

All right guys, I've been thinking and googling for a while, and I can't seem to find any source explaining why any wave refracts when entering a different medium. Same problem as before, they just seem to state that it happens, and not why it happens. This seems so incredibly basic that I'm frightened by how I'm not able to figure it out. I mean, there isn't anything "magical (i.e quantum physics)" about water and fluid dynamics! What on earth am I missing?

[Twistar]

You can understand it with just electrodynamics and conservation of momentum. Use Maxwell's equations to solve for the case of a plane wave hitting some interface (at an angle) and solve the boundary condition problem. What you will find is that the components of the wave vector need to be conserved in two directions (one parallel to the interface, and one perpendicular) but the length of the wave vector is determined by the speed of light in that medium. From all of this you can derive Snell's law. You can actually derive this for wave equations in general, it doesn't need to be light.

Wikipedia has some good pictures
The picture to the right of the "Derivations and Formulae" section is really illuminating.

[TestTubeGames]

I agree with you, Fredda, that analogies and metaphors can be spurious in physics. Necessary, but they can conceal a massive amount of hand-waving.

When it comes to waves bending, I find it helps me to think about what happens to the wave as it hits the boundary. Picture two points along a single crest of the wave. Since it is coming in at an angle, one point on the crest hits the the (let's say) slower medium first. It slows down. The other point keeps going fast for a while, until it too crosses over. This is why the wheels-in-gravel analogy comes up. If you have a line moving along, and suddenly one point on it travels faster or slower than the others, the line has to turn. Just a property of geometry. The same for waves as for cars...

Did that help?

[Fedda]

Yes, actually. I'll just adapt that anaology and think of the waves an interconnected, where the wavefronts act as geometrical lines with the associated properties. I hate to "give up", but as long as I can visualize it as well as do the math I guess it's okay. Thanks for the help, guys!

[Qaanol]

You don’t have to think of it as rigidly connected.

Imagine a whole infinite line of cows, all side by side, walking across the prairie. Let’s say there’s one cow at the “left” end of the line, and all the rest are to her right. They come, at an angle, to some muddy ground. So, there’s a long straight line of edge between solid prairie and soft muck, and the line of cows comes to it at an angle. Let’s say a 45° angle, with cow #1 getting there first.

This mud is so sticky that the cows can’t move once they get into it. And they do get to it, first the ones way on the left end of the line, then more and more as the line of cows keeps walking on its 45° angle and keeps hitting the mud. Every cow eventually hits the mud, gets stuck, and can’t move anymore.

All the cows are still facing at a 45° angle to the edge of the mud, but at the same time the line of cows now goes along the edge of the mud. So the line itself has turned 45°, even though none of the individual cows turned, they just got stuck.

Repeat, but this time the mud just slows the cows down a lot. Every cow always faces at the same original 45° angle, but once they get in the mud they are slowed down a lot. That means, as more and more cows get into the mud, the ones that are already in the mud have barely moved forward at all in the mud, but they have a little bit.

So now the line of cows in the mud makes something like a 5° angle with the edge of the mud. Each individual cow in the mud is still walking (very slowly) at the same 45° angle as ever before. But the line of cows is now at a 5° angle. If you were up in a helicopter looking down, you’d see cows in the prairie in a long line at 45° to the mud edge, and where the line of cows reaches the mud it would meet a line of cows in the mud at 5°. So you’d see a line with a single bend in it.

Now suppose the line of cows goes off infinitely in both directions, so it has no end. And suppose you’re high enough in the air that you can’t see individual cows, just the line with a bend in it. As you watch, the lines move forward. The line on the prairie moves forward, in the direction perpendicular to the 45° line itself, and the line in the mud moves forward, in the direction perpendicular to the 5° line itself.

You can’t see what the individual cows are doing, although we know they are all walking at the same 45° angle as always, but you can see what the lines are doing. And the direction of propagation of the line changes when it gets into the mud.

You can imagine this line of cows being just the very front of a huge herd, so behind the line of cows is a solid mass of more cows. Now if someone asks what direction the cows are moving, you can look down and say “The front edge of the herd is advancing at a 45° angle on the prairie, and at a 5° angle in the mud.”

[Rococo]

It looks like everyone else is happy, but none of these explanations really do it for me. Qaanol's explains the change in wavefront, but not the change in momentum direction, since in any configuration other than an infinite evenly spaced line his cows have a different group velocity and phase velocity. Explanations based on Maxwell's equations/ QM shed some nice light on these topics, but obscure the fact that diffraction doesn't require either of these things- in particular, acoustic waves obey Snell's law too. Fermat's principle is closest, but it would be nice to explain it without calculus of variations.

I don't really have a good replacement, but here's the best I can do: a pretty basic fact about a wave hitting a boundary is that it will result in both a reflected wave and a transmitted wave, with the proportion between them dependent on the difference between the media. If you haven't been shown a demo of this, youtube it. So since momentum is conserved and some of it is carried off by the reflection, the momentum of the transmitted light has to shift. If you imagine yourself as a light beam going through a barrier, and you think of how it would shift you if you 'threw' some momentum over your shoulder in the direction that a reflection should be, you can see that this pushes you onto a trajectory that is both slower and closer to the perpendicular, just like it should be.

[Fedda]

I used to think of refraction similarly to Qaanol's analogy, but I had the same problem with it as Rococo. I'm amazed that so many explanations can arise from the same observation, and the one Rococo presented is one that I have never heard, but it actually makes the most immediate sense of them all, since I can't get my head around quantum physics which seems to be the proper explanation according to the physicists. I can't stretch my amazement that so many different explanations exist

Thanks for all the answers, it does shed some light on this (ba-dumm tshh [sorry, I had to])

[starslayer]

It looks like everyone else is happy, but none of these explanations really do it for me. Qaanol's explains the change in wavefront, but not the change in momentum direction, since in any configuration other than an infinite evenly spaced line his cows have a different group velocity and phase velocity.

I'm not really sure how this is a problem, since this is exactly what happens to light in any real medium (it's not even a problem for either or both of them to be greater than c, either). All the cows are is a square wave, really, and square waves are perfectly valid solutions to the wave equation.

With the explanations based on Maxwell's equations, it should be fairly obvious that they generalize almost immediately, since the whole idea that light is an electromagnetic wave is explicitly predicated on the fact that Maxwell's equations immediately give you the wave equation if assume a region with no charge or current density (i.e., free space), and it only takes more massaging to get wave equations in the presence of charge and current.

Also, if you throw momentum over your shoulder (I'm assuming you mean opposite your direction of travel), you don't slow down. You speed up. You do move closer to the perpendicular in the reflection/transmission case, and it is a nice illustration of why the transmitted wave's direction changes, but you need the fact that real media are intrinsically absorptive to account for the missing forward momentum.

since I can't get my head around quantum physics which seems to be the proper explanation according to the physicists

This is the ultimate explanation, sure, but you don't need QM to understand and explain refraction. Refraction, reflection, diffraction, and all the rest are properties of all waves, classical and quantum mechanical.

[Twistar]

Ok, don't worry about quantum mechanics or electrodynamics. It's just the wave equation partial differential equation solved on boundary conditions. These things are waves so they satisfy the wave equation in each of the two different media. What this amounts to is that the waves have different velocity (or k vectors) on each side of the boundary. (If you want to know why the different materials change the speeds of the wave that's a different question but I don't think anyone is asking that). Now, just write down the equations for an incident, reflected, and transmitted wave, all with the same frequency and you can derive Snell's law from the fact that all of these waves must agree at all points and times along the boundary.

I think the trick is that since the waves all have the same frequency but different velocities something has to give to allow them to be equal on the boundary. Imagine if diffraction didn't happen. You would have one wave entering the boundary at an angle and a transmitted wave leaving at the same angle. Both of these waves have the same frequency but the one wave is moving faster. You would immediately get a disagreement between the two waves. wave fronts would crash into the boundary faster than they left it. The angle changes to compensate for the phase mismatch.