Why Light Slows Down: Feynman's Refractive Index Explained — Full Breakdown & Transcript

[00:00] The standard explanation, what you might hear in a high school physics class for example, When light enters a medium, like glass, it slows down,

[00:13] in a vacuum those crests are traveling at c, the speed of light, but inside the glass those crests will be traveling a little bit slower. And the specific ratio between the speed of light in a vacuum and the speed

[00:29] inside a medium like this is called the index of refraction for that medium. The reason we use the word refraction instead of say the index of slowing is that if a beam of light enters this glass at an angle,

[00:42] then a consequence of this slowdown is that it bends a little bit, And the way my high school physics teacher always explained this was to imagine a tank going from some region where it can travel relatively quickly, like concrete,

[00:56] into something slower, like mud, where if it's coming in at an angle, that tread will be going slower while the other one is faster, until that second tread also enters the mud, then it continues straight,

[01:12] We'll get back to the actual reason for bending in a bit, but at this point the high school physics students typically learn a law known as Snell's law, which specifies exactly how much things bend.

[01:25] If you draw a line perpendicular to the boundary between the glass and water, and consider the angle between that perpendicular line and the beam of light, then Snell's law tells us that the sine of this angle divided by the speed of the light

[01:38] So the slower the light, the lower that angle will be, and that lets you actually calculate how much things refract. that light slows down depends a little bit on its frequency.

[01:53] For example, blue light, which has a relatively high frequency, which has a relatively low frequency. Now most of the light that you see is not a clean pure sine wave,

[02:06] in particular the white light coming from the sun is not a clean sine wave, it's something much messier, but it can be expressed as a sum of a bunch of clean sine waves, each one corresponding to a pure spectral color.

[02:18] all those different components get refracted by slightly different amounts, causing this iconic separation of the pure rainbow colors. So that is the standard explanation, and it's not wrong per se,

[02:32] it's just that all of the key components are handed down from on high. And what exactly do we mean by slowing down? slows down have anything to do with the color of the light?

[02:47] If you have a sufficiently high standard for explanations, rather than feeling like they were handed down. The first explanation I saw that started to give this feeling came from

[03:01] this video is simply animate a lot of the key points that he makes there. charge in the material, and the propagating light waves caused by each

[03:15] one of those charges, and how all of them superimpose on top of each other. but it actually works out to be not only understandable, but satisfyingly explanatory. and the key intuition there really comes down to what happens if you're bad at

[03:33] Bear with me, I promise that'll make sense later. a lot of people had a lot of questions about the index of refraction. For example, numerous people asked about how it's possible for this number

[03:48] despite that seeming to imply the impossibility of something traveling There was also a question about birefringence, causing you to c-double when you look through it.

[04:05] piece from the last two videos about the barber pole phenomenon. And a couple people also asked about why light slowing down would imply a bending like this, and I agree that deserves a better explanation than the tank analogy.

[04:21] but it makes sense to first lay down some groundwork by spending the bulk of our time on the key question of why passing through a medium would change the And for this, I want you to think of your material, like glass,

[04:37] all perpendicular to the direction the light is traveling. effect of just one of those layers on the light wave.

[04:49] The true effect would be miniscule, but if you'll let me exaggerate it for a moment, what it does is kick back the phase of the wave. all on the same page when it comes to wave terminology.

[05:01] If you go and graph the function sine of x, when you put some term in front of it, affecting how high that wave oscillates up and down, that's what we call the amplitude, when you put a term in front of x, this will affect how rapidly it oscillates.

[05:14] that term would be called the angular frequency, that constant would be called the wave number. Then if you were to add some other constant inside that sine function,

[05:28] it sort of slides the wave left and right, that term describes the phase of the wave. So when I say that our light wave hitting a layer of glass causes its phase to

[05:40] get kicked back, I mean if you take whatever function describes it before it hits the glass, then the function describing it after that looks almost the same, just with a little extra something added to the input of that sine function.

[05:52] something proportional to the infinitesimal thickness of that layer, but I'll keep drawing it as something exaggerated and keep track of the value of Let's say you go and add a bunch of other layers of the glass,

[06:07] each one also applying their own kickback to the phase of the wave. If the value of that phase kick applied by each layer is something really close to zero, But the larger that phase kick, the more the wave kind

[06:23] Admittedly, right here it looks all kaleidoscopic and weird, but that's really just because I have a discrete set of layers, Notice what happens if I smooth it out by doubling the density of layers,

[06:39] but having each one only apply half the phase kick. but have each one only apply half the phase kick. As I continue this over and over, approaching a situation where you have a continuum of

[06:54] glass, each layer applying just a tiny infinitesimal phase kick, what you end up with is identical to, indistinguishable from, a wave that's simply traveling slower, oscillating up and down with the same frequency,

[07:07] This right here is the first key idea with the index of refraction. what we really need to ask is why does its interaction with

[07:21] a single layer of that glass cause a kickback to the phase of the wave? And then when we want to get quantitative and understand exactly how much the light slows down, which is critical for understanding why it depends on color,

[07:34] instead the real question is how strong is that phase kick? From here it's helpful to turn back to the fundamentals of what light even is. but a little review never hurts so let me go over the essentials.

[07:48] As many of you know, light is a wave in the electromagnetic field, The electric field associates each point in 3D space with a little 3-dimensional vector telling you what force would be applied to

[08:03] a hypothetical unit charge sitting at that point in space. The key thing going on with light is that if you have a charged particle that results in these propagating ripples in the electric field away from the charge,

[08:19] and that propagation is traveling at the speed c, the speed of light. they cause it to wiggle up and down, albeit a little more weakly than the initial wiggle, and that in turn causes its own propagations.

[08:34] The way we described this in the last video was that if at some point in time a charge is accelerating, then after a little delay, which depends on this speed c, the existence of that acceleration induces a force on another charge.

[08:48] it's something that can be derived downstream of Maxwell's equations, but for our purposes here, the main thing to tuck away in your mind is that the amount of time it takes that initial acceleration to cause any kind of influence

[09:02] And really, you should think of c not so much as the speed of light per se, It determines how fast any kind of influence travels, it's just that one of multiple consequences of that is that it's the speed of light.

[09:18] In particular, when you get a charge oscillating up and down in a nice clean sinusoidal motion, you can think of these rippling effects in another charge sitting there as a result of that past acceleration.

[09:34] I will freely admit that I had a bit too much fun in that video just simulating how the and that I'm kind of doing the same thing here, but there are two important facts for our pursuit of the index of refraction.

[09:46] The first is that when you have multiple different charges oscillating up and down, for each individual charge, which is kind of what you would expect.

[09:58] And then the way that it shakes out is that if you have a row of charges oscillating in sync with each other, or for our purposes today, a plane of charges all wiggling up and down in sync within that plane,

[10:10] then the effects of each individual charge tend to cancel each other out in they actually constructively interfere. That's the important thing.

[10:24] If you have a layer of charges wiggling up and down in sync with each other, then even far away from that layer, it produces this nice sinusoidal wave in the electric field that we're so fond of drawing to represent light.

[10:37] depicting the electric field on a single one-dimensional line. A more full picture of light in three dimensions would look something more like this. That tends to be a little bit busier, so usually we just draw the sine wave.

[10:52] So thinking back to the question of why interactions with a layer of material would cause a kickback to the phase of the wave, let's start thinking it through. then it causes all of the charges inside that material, you know,

[11:08] electrons or maybe the occasional ion, to wiggle up and down in response You might think that adding together all the propagations from all those charges is a complete nightmare, but we can think about it one layer at a time.

[11:22] that wiggling produces its own second-order light wave at the same frequency. And it propagates in both directions perpendicular to that layer.

[11:34] incoming light wave added together with the second-order wave. By far the most distracting part of what's going on here is everything on the left,

[11:46] and this actually corresponds to the light being reflected back. light not only goes through it, but some of it gets reflected back. And we could have a whole interesting discussion on quantifying exactly how much,

[12:00] but in the spirit of staying focused, we will completely ignore that for today and only focus on what's happening to the right of that layer. It turns out that when you add that second-order oscillation,

[12:14] but just shifted back in phase by a little bit. And then because many successive shifts to the phase like this are the same thing as light slowing down, this will ultimately explain the index of refraction.

[12:29] hands and asking, why is that the effect when you add them together? on how to think about adding two waves together. If you draw some sine wave with some particular amplitude, some specific frequency,

[12:45] also with its own amplitude, frequency, and phase, in general it's very hard to think about what the sum of those two waves should look like as you tweak those initial parameters.

[13:02] In the specific case where the frequencies are the same, which is true for our example, the result will also look like a sine wave with that same frequency. But even then it's a little tricky to think about exactly how to describe that wave.

[13:15] It has some amplitude and some phase, and if I ask you to concretely compute both of those numbers based on the amplitudes and phases of the initial waves, it's not immediately clear how you would do that without throwing a bunch of trig

[13:28] But here's a really nice way to think about it. Imagine that first wave describes the y component of some rotating vector. The length of that vector corresponds with the amplitude of our wave,

[13:42] and then the initial rotation of that vector corresponds with the phase of our wave. Similarly, think of that second wave as describing the y component of another rotating vector, where again the amplitude corresponds with the length of that vector,

[13:56] and the phase of the wave tells us the initial angle of that vector. think about adding those two vectors tip to tail.

[14:08] And because they both have the same frequency as both of them rotate, their sum rotates in lockstep with them. it comes down to the length of this vector sum,

[14:23] and similarly the phase corresponds to the angle of that vector sum. like if the two phases happen to be the same, then you get constructive interference and you have a bigger wave that results.

[14:36] then you get deconstructive interference with a relatively small resulting wave. What's a little bit less obvious, but what's crucial for our discussion here,

[14:48] is that if the phase of that second wave happens to be exactly 90 degrees behind the phase of the first, so kind of a quarter cycle out of sync, then if you look at the little vector sum on the lower left,

[15:03] you'll notice how this means that the resulting wave is almost identical to the initial wave, but just shifted back in its phase by a tiny bit. on the specific amplitude of that second wave.

[15:18] So looking back at our previous animation, where we have some wiggling charges in a layer of glass causing these second order propagations that need to be added together with the incoming light, the way it works out is that the phase

[15:31] of that second wave is exactly a quarter of a cycle behind the phase of the first. And then, critically, the size of that phase shift is bigger when that second

[15:43] order wave is larger, and then smaller when that second order wave is smaller. why does it work out to be exactly a quarter of a cycle behind?

[15:56] There is a very nice reason, but it's just a little too much detail for us today. a look at the Feynman lectures on the matter. For our purposes, step back for a second and think about what you need to explain the key

[16:09] question of prisms, which is why the index of refraction would depend on color at all. As you now know, that index depends on how much each layer of glass kicks back the phase of the wave, and that phase kick depends on the strength of the

[16:23] second order wave resulting from charge oscillations in a layer of that glass. those charges wiggle in response to an incoming light wave.

[16:35] So let's zoom in on that layer and think of each one of those charged particles, and even though the specific molecular structure is going to be something very complicated, we're going to model each one of those charges as if it

[16:47] was bound to some equilibrium position by a spring, or maybe a set of springs. I don't mean this literally, of course, I just mean if we describe the displacement of this charge from its equilibrium with a little vector x

[16:59] that's going to depend on time, then in our model, the force applied to the charge, pulling it back to that equilibrium, is going to be something proportional to the size of that displacement, with a little proportionality constant k.

[17:13] You might ask if that's accurate, and the idea is that for very small displacements, This is a very common thing to do throughout physics, The idea is that maybe the actual force law depends on the position in a much more

[17:30] complicated way, but we're basically taking a low order approximation near the If I just run this as a simulation, plugging in this force law, here's what that displacement looks like as a function of time.

[17:42] What you get looks like a sine wave, this is called simple harmonic motion, and the frequency of this wave is going to matter a lot for you and me, because the force is really the same thing as mass times acceleration,

[17:57] and the acceleration is the same thing as the second derivative of that displacement. derivative looks like a certain constant times that function itself.

[18:09] enjoy thinking about how you solve this. and anyone who knows a little calculus can just check it for themselves. The way it shakes out is that if the initial condition is that our little charge has a

[18:24] velocity of zero, but it's offset from the equilibrium by a little vector x-naught, then the way it evolves over time looks like x-naught multiplied by a cosine expression. it just depends on how far we pulled things back originally,

[18:41] but the meet is this frequency term, square root of k divided by m. For example, if you increase k, which is kind of like increasing the strength of that spring, then it results in a faster oscillation.

[18:57] there's a lot more inertia and it results in a slower oscillation. it's called the resonant frequency for our simple harmonic oscillator.

[19:11] And being a little more precise, I should call this the resonant angular frequency. where whenever you have some kind of cyclic process, when you give an intuitive description, it's natural to phrase things in terms of the

[19:24] But when doing math, it's often more natural to talk about the angular frequency, process covers in radians per unit time.

[19:36] So for example, if you have something like a cosine expression, which you might think of as describing the x component of a cycling vector like this, then the term sitting right in front of the t in that cosine is the angular frequency.

[19:51] For example, in our simple harmonic motion, the term sitting in front of t looks like the square root of k divided by m, which I'm writing as omega sub r. case where there's no external force acting on our charged particle.

[20:09] But of course, what we're interested in is what happens when causes this charge to jiggle, but the question is how much. In our equation, this looks like adding a new force term corresponding to the light wave.

[20:25] That force oscillates up and down, also according to some kind of cosine function, but this time with a distinct angular frequency that I'm going to call omega sub l. and then q describes the charge of whatever particle we're modeling.

[20:41] As usual, it's a lot easier to think about when we only draw a subset of that light wave, layer of material we care about. on the spring up and down in a clean sinusoidal pattern.

[20:57] Or as another analogy, it's similar to pushing a child on a swing. The swing would oscillate on its own due to the force of gravity, but you as the pusher are applying an external force which itself is oscillating

[21:09] Although a key difference here is that the frequency of that external force in general has nothing to do with the resonant frequency of that little oscillator. The better analogy would be if you're pushing the child on the swing with a

[21:24] cyclic force that has nothing to do with what the swing naturally wants to do. And my favorite part in literally trying to do this with my niece is that at some point she gently murmurs to herself, this isn't how mom does it.

[21:37] Now, in trying to understand how much our charge is oscillating in response to the incoming light, let me start by just simulating it and plotting the result. You'll notice that there's a little startup period where it kind of has to get going,

[21:51] but then after that, mercifully, it looks nice and clean, just like another sine wave. but it's important to understand that this one has a very different character from the

[22:03] Earlier, without any external forces, the frequency of that wave came down to the spring constant and the mass, which is to say, it depends exclusively on material properties of the glass.

[22:17] the frequency in that steady state is the same as the frequency of the light. it just depends on how far you pulled the spring out to begin with.

[22:32] is actually where all the interesting stuff happens. Exactly how much will this charge be oscillating in response to the light wave? but any eager calculus students among you might enjoy going through the exercise where

[22:51] if you just guess that a solution looks like a cosine wave with the same frequency as the light, and you solve for the amplitude, you can get a concrete solution to this This is worth unpacking for a bit, and just to be clear,

[23:05] after things have gotten up and going. A fully descriptive solution would be notably more complicated. which here looks like a large collection of constants,

[23:20] most of which should be pretty intuitive if you take a moment to think about it. so the stronger the light the more the oscillations. It's also proportional to the charge, which again makes sense.

[23:34] And the real heart of the matter comes down to what's sitting in the denominator here, and the square of the light frequency. what would happen if the frequency of the incoming light was

[23:49] something very close to the resonant frequency of this oscillator. where the frequency of your force lines up quite closely with what the swing wants to do.

[24:03] In this case, running the simulation, notice how the oscillations of that particle will grow and grow and grow, becoming quite large over time. Some of you may know the famous example of the Millennium Bridge in London,

[24:17] where on its opening day it started oscillating way more than the engineers expected And what was going on is that the frequency of the steps of the crowd lined up very closely with a resonant frequency, causing this worryingly high amplitude.

[24:33] By contrast, notice what happens in the simulation if the frequency of the light, ωL, is something much smaller than the resonant frequency. things get into their full swing, eventually it finds a nice sinusoidal motion,

[24:50] but the amplitude of that motion is much more modest in comparison. So what our equation is telling us is that the larger the difference between those so the smaller the overall wiggle to that charge.

[25:04] As I'm applying a force with a frequency that's very different from what the swing wants to she ends up oscillating at the same frequency as my force,

[25:16] but she's going at a relatively low amplitude. Stepping back, what this means is that as you shine light into a material like glass, it's not just that it induces wiggles in the charges of that material,

[25:29] but the specific size of those wiggles depends on the frequency of And the more those wiggle, the bigger the size of this second order wave caused by

[25:41] that layer, which in turn causes a bigger shift to the phase of the overall wave. apparent slowdown to the light, it means that the amount that it

[25:53] will slow down ultimately depends on the frequency of the light. You cannot truly explain the light separation until you get down to the driven harmonic oscillator.

[26:07] I encourage the curious viewers to take a look at the Feynman lectures that One quite important detail that would be a little criminal not to mention is that when

[26:19] force, there should really also be a term that depends on the velocity of that charge. This term accounts for the fact that energy from

[26:32] Without it, this whole explanation would seem to imply that light always passes through every material, not just glass and water, when as you can tell just by looking around,

[26:44] there's all sorts of materials for which light is mostly reflected and absorbed. As I mentioned at the start, folks on Patreon had numerous questions and why slowing implies bending, so I made a supplemental video answering

[26:59] a handful of those questions, which should be published in just a few days. In the meantime, my friend Mithena from the channel Looking Glass Universe just put out a pair of videos on the related but definitely distinct question

[27:11] crests of a clean, pure sine wave in a steady state, but in the sense of trying to send information through that medium, I definitely owe the existence of this video to many conversations with her about this

[27:27] especially at the second one. By the way, some collaborators and I made this notebook that I think a lot of viewers might enjoy, and given that it's the holiday season it seems worth a quick mention.

[27:41] The premise is that every one of the pages has a quote that's related to math, to quotes conveying some genuinely thought-provoking idea. And then aside from the content, I basically made the kind of notebook that I most

[27:56] enjoy taking notes in, something that's readily portable with very faint gridlines all bound in this nice soft faux leather. store next to a lot of other mathematical merchandise.